# Programs for A100000-A149999

List of integer sequences with links to LODA programs.

- A100005 (program): Bisection of A001414.
- A100006 (program): Integer log of 2n: sum of primes dividing 2n (with repetition).
- A100007 (program): Number of unitary divisors of 2n-1 (d such that d divides 2n-1, GCD(d,(2n-1)/d)=1). Bisection of A034444.
- A100008 (program): Number of unitary divisors of 2n.
- A100019 (program): a(n) = n^4 + n^3 + n^2.
- A100021 (program): Numbers of the form 3prime(n) - prime(n+1) - 3.
- A100029 (program): Bisection of A008472.
- A100030 (program): Bisection of A008472.
- A100031 (program): Bisection of A005384.
- A100032 (program): Bisection of A005384.
- A100033 (program): Bisection of A001700.
- A100036 (program): a(n) = smallest m such that A100035(m) = n.
- A100037 (program): Positions of occurrences of the natural numbers as a second subsequence in A100035.
- A100038 (program): Positions of occurrences of the natural numbers as third subsequence in A100035.
- A100039 (program): Positions of occurrences of the natural numbers as fourth subsequence in A100035.
- A100040 (program): a(n) = 2*n^2 + n - 5.
- A100041 (program): a(n) = 2*n^2 + n - 7.
- A100042 (program): a(n) = prime(n)*2^prime(n).
- A100043 (program): a(n) = (3*n-1)!.
- A100044 (program): Decimal expansion of Pi^2/9.
- A100047 (program): A Chebyshev transform of the Fibonacci numbers.
- A100048 (program): A Chebyshev transform of the Pell numbers.
- A100049 (program): A Chebyshev transform of the Padovan numbers.
- A100050 (program): A Chebyshev transform of n.
- A100051 (program): A Chebyshev transform of 1,1,1,…
- A100052 (program): A Chebyshev transform of the odd numbers.
- A100053 (program): Maximum run of white (or OFF) cells in generation n of the Rule 30 elementary cellular automaton.
- A100057 (program): Sum of absolute differences of p(n) defined in A054065, oriented around a clock.
- A100058 (program): Expansion of 1 / (1 - 3x - x^2 + 2x^3).
- A100059 (program): First differences of A052911.
- A100061 (program): Numerator of the probability that an integer n occurs in the cumulative sums of the decimal digits of a random real number between 0 and 1.
- A100062 (program): Denominator of the probability that an integer n occurs in the cumulative sums of the decimal digits of a random real number between 0 and 1.
- A100063 (program): A Chebyshev transform of Jacobsthal numbers.
- A100066 (program): Expansion of x/((1-x)sqrt(1-4x^2)).
- A100067 (program): a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*2^(n-2*k).
- A100068 (program): a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*3^(n-2*k).
- A100069 (program): a(n) = Sum_{k=0..floor(n/2)} binomial(n,k)*4^(n-2*k).
- A100070 (program): Number a(n) of forests with two components in the complete bipartite graph K_{n,n}.
- A100071 (program): a(n) = n * binomial(n-1, floor((n-1)/2)) = n * max_{i=0..n} binomial(n-1, i).
- A100073 (program): Number of representations of n as the difference of two positive squares.
- A100087 (program): Expansion of x/(sqrt(1-4*x^2) + x - 1).
- A100088 (program): Expansion of (1-x^2)/((1-2*x)*(1+x^2)).
- A100089 (program): a(n) = (3*n+1)!.
- A100095 (program): An inverse Chebyshev transform of the Fibonacci numbers.
- A100096 (program): An inverse Chebyshev transform of the Jacobsthal numbers.
- A100097 (program): An inverse Chebyshev transform of the Pell numbers.
- A100098 (program): An inverse Chebyshev transform of (1-x)/(1-2x).
- A100099 (program): An inverse Chebyshev transform of x/(1-2x).
- A100100 (program): Triangle T(n,k) = binomial(2*n-k-1, n-k) read by rows.
- A100101 (program): Bell(2n)*(2n-1)!!, where Bell are the Bell numbers A000110.
- A100102 (program): a(n) = 2^(2*n)-(2*n-1).
- A100103 (program): a(n) = 2^(2*n) - 2*n.
- A100104 (program): a(n) = n^3 - n^2 + 1.
- A100105 (program): a(n) = 2^prime(n)-prime(n).
- A100107 (program): Inverse Moebius transform of Lucas numbers (A000032) 1,3,4,7,11,..
- A100109 (program): a(n) = n^3 - 2*n^2 + 2.
- A100111 (program): a(n) = Sum_{k >= 0} prime(n-4k).
- A100112 (program): If n is the k-th squarefree number then a(n) = k, otherwise a(n) = 0.
- A100118 (program): Numbers whose sum of prime factors is prime (counted with multiplicity).
- A100119 (program): a(n) = n-th centered n-gonal number.
- A100130 (program): Expansion of (eta(q) * eta(q^4) / eta(q^2)^2)^24 in powers of q.
- A100131 (program): a(n) = Sum_{k=0..floor(n/4)} binomial(n-2k, 2k)*2^(n-4k).
- A100132 (program): a(n) = Sum_{k=0..floor(n/4)} C(n-2k,2k) * 2^(n-3k).
- A100133 (program): a(n) = Sum_{k=0..floor(n/4)} C(n-2k,2k) * 3^k * 2^(n-4k).
- A100134 (program): a(n) = Sum_{k=0..floor(n/6)} binomial(n-3k,3k).
- A100135 (program): a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 2^k.
- A100136 (program): a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 3^k.
- A100137 (program): a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 2^(n-6k).
- A100138 (program): a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 2^(n-5k).
- A100139 (program): a(n) = Sum_{k=0..floor(n/6)} C(n-3k,3k) * 3^k * 2^(n-6k).
- A100143 (program): Unique sequence with a(1)=1 where each a(n) occurs in the same order a(n) times consecutively in its sequence of first differences which contains no other terms.
- A100144 (program): First differences of A100143.
- A100145 (program): Structured great rhombicosidodecahedral numbers.
- A100146 (program): Structured great rhombicubeoctahedral numbers.
- A100147 (program): Structured icosidodecahedral numbers.
- A100148 (program): Structured small rhombicosidodecahedral numbers.
- A100149 (program): Structured small rhombicubeoctahedral numbers.
- A100150 (program): Structured snub cubic numbers.
- A100151 (program): Structured snub dodecahedral numbers.
- A100152 (program): Structured truncated cubic numbers.
- A100153 (program): Structured truncated dodecahedral numbers.
- A100154 (program): Structured truncated icosahedral numbers.
- A100155 (program): Structured truncated octahedral numbers.
- A100156 (program): Structured truncated tetrahedral numbers.
- A100157 (program): Structured rhombic dodecahedral numbers (vertex structure 9).
- A100158 (program): Structured disdyakis triacontahedral numbers (vertex structure 11).
- A100159 (program): Structured disdyakis triacontahedral numbers (vertex structure 7).
- A100160 (program): Structured disdyakis triacontahedral numbers (vertex structure 5).
- A100161 (program): Structured disdyakis dodecahedral numbers (vertex structure 9).
- A100162 (program): Structured disdyakis dodecahedral numbers (vertex structure 7).
- A100163 (program): Structured disdyakis dodecahedral numbers (vertex structure 5).
- A100164 (program): Structured rhombic triacontahedral numbers (vertex structure 11).
- A100165 (program): Structured rhombic triacontahedral numbers (vertex structure 7).
- A100166 (program): Structured deltoidal hexacontahedral numbers (vertex structure 9).
- A100167 (program): Structured pentagonal icositetrahedral numbers (vertex structure 13).
- A100168 (program): Structured pentagonal icositetrahedral numbers (vertex structure 10).
- A100169 (program): Structured pentagonal hexacontahedral numbers (vertex structure 16).
- A100170 (program): Structured pentagonal hexacontahedral numbers (vertex structure 10).
- A100171 (program): Structured triakis octahedral numbers (vertex structure 4).
- A100172 (program): Structured triakis icosahedral numbers (vertex structure 4).
- A100173 (program): Structured pentakis dodecahedral numbers (vertex structure 6).
- A100174 (program): Structured tetrakis hexahedral numbers (vertex structure 5).
- A100175 (program): Structured triakis tetrahedral numbers (vertex structure 4).
- A100176 (program): Structured octagonal prism numbers.
- A100177 (program): Structured meta-prism numbers, the n-th number from a structured n-gonal prism number sequence.
- A100178 (program): Structured hexagonal diamond numbers (vertex structure 5).
- A100179 (program): Structured heptagonal diamond numbers (vertex structure 5).
- A100181 (program): Odd terms in A120070.
- A100182 (program): Structured tetragonal anti-prism numbers.
- A100183 (program): Structured hexagonal anti-prism numbers.
- A100184 (program): Structured octagonal anti-prism numbers.
- A100185 (program): Structured meta-anti-prism numbers, the n-th number from a structured n-gonal anti-prism number sequence.
- A100186 (program): Structured heptagonal anti-diamond numbers (vertex structure 7).
- A100187 (program): Structured octagonal anti-diamond numbers (vertex structure 7).
- A100188 (program): Polar structured meta-anti-diamond numbers, the n-th number from a polar structured n-gonal anti-diamond number sequence.
- A100189 (program): Equatorial structured meta-anti-diamond numbers, the n-th number from an equatorial structured n-gonal anti-diamond number sequence.
- A100190 (program): The (4,1)-entry in the 4 X 4 matrix M^n, where M = [1,0,0,0 / 3,3,0,0 / 3,6,3,0 / 1,3,3,1].
- A100191 (program): The (1,1)-entry in the 3 X 3 matrix M^n, where M = [1,2,1 / 2,2,0 / 1,0,0].
- A100192 (program): a(n) = Sum_{k=0..n} binomial(2n,n+k)*2^k.
- A100193 (program): a(n) = Sum_{k=0..n} binomial(2n,n+k)*3^k.
- A100196 (program): Numbers of positive integer cubes <= n^2.
- A100197 (program): Numbers of squares in the range [n^3, (n+1)^3].
- A100198 (program): Let f(0) = -1, f(n) = Moebius(n) = A008683(n) for n>0. Sequence gives partial sums a(n) = Sum_{ 0 <= i <= n} f(i).
- A100201 (program): Primes of the form 23n+3.
- A100202 (program): Primes of the form 13*k + 3.
- A100203 (program): Primes of the form 37n+3.
- A100204 (program): Numbers of cubes in the range [n^2, (n+1)^2].
- A100206 (program): Row sums of Clark’s triangle A046902.
- A100207 (program): a(n) = 4 + 8*n + 10*n^2 + 4*n^3.
- A100212 (program): Expansion of (x^5 + 2*x^4)/(1/2*x^2 - 2*x^6 + 2*x^5 - x^4 - 1/2*x + 1/4).
- A100213 (program): G.f. x* (4-7*x+2*x^2-8*x^4+16*x^5-16*x^6) / ((2*x-1) * (2*x^2-1) * (2*x^2-2*x+1) * (2*x^2+1)).
- A100214 (program): a(n) = 4*n^3 + 4.
- A100215 (program): Expansion of (4 - 7*x + 2*x^2)/((1-2*x)*(1 - 2*x + 2*x^2)).
- A100216 (program): Relates row sums of Pascal’s triangle to expansion of cos(x)/exp(x).
- A100217 (program): Diagonal sums of a binomial number triangle.
- A100218 (program): Riordan array ((1-2x)/(1-x), (1-x)).
- A100219 (program): Expansion of (1-2*x)/((1-x)*(1-x+x^2)).
- A100223 (program): G.f. A(x) satisfies: 2^n - 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (2+z)^n - (1+z)^n + z^n = Sum_{k=0..n} x^k^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.
- A100224 (program): Triangle, read by rows, of the coefficients of [x^k] in G100224(x)^n such that the row sums are 2^n-1 for n>0, where G100224(x) is the g.f. of A100224.
- A100225 (program): G.f. A(x) satisfies: 3^n - 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (3+z)^n - (1+z)^n + z^n = Sum_{k=0..n} x^k^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.
- A100226 (program): Triangle, read by rows, of the coefficients of [x^k] in G100225(x)^n such that the row sums are 3^n-1 for n>0, where G100225(x) is the g.f. of A100225.
- A100227 (program): Main diagonal of triangle A100226.
- A100230 (program): Main diagonal of triangle A100229.
- A100233 (program): a(n) = Lucas(3*n) - 1.
- A100236 (program): Main diagonal of triangle A100235.
- A100237 (program): Secondary diagonal of triangle A100235 divided by row number: a(n) = A100235(n+1,n)/(n+1) for n >= 0.
- A100238 (program): G.f. A(x) satisfies: 2^n + 1 = Sum_{k=0..n} [x^k] A(x)^n for n>=1.
- A100239 (program): G.f. A(x) satisfies: 3^n + 1 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: (3+z)^n + (1+z)^n - z^n = Sum_{k=0..n} x^k^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.
- A100240 (program): G.f. A(x) satisfies: 4^n/2 = Sum_{k=0..n} [x^k]A(x)^n and also satisfies: ((4+z)^n + z^n)/2 = Sum_{k=0..n} x^k^n for all z, where [x^k]A(x)^n denotes the coefficient of x^k in A(x)^n.
- A100242 (program): a(n) = n^5 - n^2*(n^2 - 1)/2.
- A100244 (program): a(n) = smallest positive integer such that {1 + product{k=1 to n} a(k)} is coprime to n.
- A100247 (program): Slanted Catalan convolution table, read by rows of 2*n+1 terms in row n, where T(n,k) = C(n+2*k-[k/2],k)*(n-[k/2])/(n+2*k-[k/2]).
- A100248 (program): Row sums of the slanted Catalan convolution table A100247.
- A100249 (program): Antidiagonal sums of the slanted Catalan convolution table A100247.
- A100250 (program): Positions where values change in A100144.
- A100255 (program): Squares of pentagonal numbers: a(n) = (1/4)*n^2*(3*n-1)^2.
- A100256 (program): Squares of second pentagonal numbers: (1/4) n^2(3n+1)^2.
- A100257 (program): Triangle of expansions of 2^(k-1)*x^k in terms of T(n,x), in descending degrees n of T, with T the Chebyshev polynomials.
- A100259 (program): Coefficient of x^2 in 2n-th normalized Legendre polynomial.
- A100260 (program): Fixed point of morphism 0 -> 01, 1 -> 02, 2 -> 31, 3 -> 32.
- A100262 (program): Expansion of A(x)^2, where A(x) = o.g.f. of n^n (A000312).
- A100279 (program): a(n) = A100107(A000032(n)).
- A100280 (program): Inverse permutation to A099896.
- A100282 (program): a(n) = A100280(A100280(n)).
- A100283 (program): a(n) = floor(p*(n+1)) - floor(p*(n)) - 1 where p = Padovan plastic number = 1.324718… (cf. A060006).
- A100284 (program): Expansion of (1-4x-x^2)/((1-x)(1-4x-5x^2)).
- A100285 (program): Expansion of (1+5x^2)/(1-x+x^2-x^3).
- A100286 (program): Expansion of (1+2x^2-2x^3+2x^4)/(1-x+x^2-x^3+x^4-x^5).
- A100287 (program): First occurrence of n in A100002; the least k such that A100002(k) = n.
- A100290 (program): Numbers divisible by smallest number with same number of 1’s in its binary expansion. That is, A038573(a(n)) divides a(n).
- A100295 (program): Simple recursive sequence generated from a symmetric matrix.
- A100296 (program): Sequence generated from a symmetric matrix.
- A100299 (program): Number of dissections of a convex n-gon by nonintersecting diagonals into an even number of regions.
- A100300 (program): Number of dissections of a convex n-gon by nonintersecting diagonals into an odd number of regions.
- A100302 (program): Expansion of (1 - x - 6*x^2)/((1 - x)*(1 - x - 8*x^2)).
- A100303 (program): Expansion of (1 - x - 4*x^2)/(1 - x - 8*x^2).
- A100304 (program): Expansion of (1 - x - 6*x^2)/(1 - x - 8*x^2).
- A100305 (program): Expansion of (1 - x - 4*x^2)/(1 - 2*x - 7*x^2 + 8*x^3).
- A100307 (program): Modulo 2 binomial transform of 3^n.
- A100308 (program): Modulo 2 binomial transform of 5^n.
- A100309 (program): Modulo 2 binomial transform of 6^n.
- A100310 (program): Modulo 2 binomial transform of 7^n.
- A100311 (program): Modulo 2 binomial transform of 8^n.
- A100312 (program): Number of 3 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (10;0) and (01;1).
- A100313 (program): Number of 4 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (10;0) and (01;1).
- A100314 (program): Number of 2 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).
- A100315 (program): Number of 3 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).
- A100316 (program): Number of 4 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).
- A100317 (program): Numbers k such that exactly one of k - 1 and k + 1 is prime.
- A100318 (program): Numbers n such that at least one of n-1 and n+1 is composite.
- A100319 (program): Even numbers m such that at least one of m-1 and m+1 is composite.
- A100320 (program): A Catalan transform of (1 + 2*x)/(1 - 2*x).
- A100321 (program): The trinomial transform (A027907) gives powers of 2, while the trinomial transform of this sequence shift one place left gives powers of 3.
- A100326 (program): Triangle, read by rows, where row n equals the inverse binomial of column n of square array A100324, which lists the self-convolutions of SHIFT(A003169).
- A100327 (program): Row sums of triangle A100326, in which row n equals the inverse binomial of column n of square array A100324.
- A100328 (program): Column 1 of triangle A100326, in which row n equals the inverse binomial of column n of square array A100324, with leading zero omitted.
- A100329 (program): a(n) = -a(n-1)-a(n-2)-a(n-3)+a(n-4), a(0)=0, a(1)=1, a(2)=-1, a(3)=0.
- A100334 (program): An inverse Catalan transform of F(2n).
- A100335 (program): An inverse Catalan transform of J(2n).
- A100336 (program): Arshon’s sequence with a different start: start from 2 and replace the letters in odd positions using 1 -> 123, 2 -> 231, 3 -> 312 and the letters in even positions using 1 -> 321, 2-> 132, 3 -> 213.
- A100337 (program): Arshon’s sequence with a different start: start from 3 and replace the letters in odd positions using 1 -> 123, 2 -> 231, 3 -> 312 and the letters in even positions using 1 -> 321, 2-> 132, 3 -> 213.
- A100340 (program): Numerators of the convergents in the continued fraction expansion for the constant given by A100338, where the partial quotients equal A006519 (greatest power of 2 dividing n).
- A100341 (program): Denominators of the convergents in the continued fraction expansion for the constant given by A100338, where the partial quotients equal A006519 (greatest power of 2 dividing n).
- A100342 (program): Numerators of the convergents in the continued fraction expansion for twice the constant given by A100338, where the partial quotients equal A006519 (greatest power of 2 dividing n) interleaved with 2’s.
- A100343 (program): Denominators of the convergents in the continued fraction expansion for twice the constant given by A100338, where the partial quotients equal A006519 (greatest power of 2 dividing n) interleaved with 2’s.
- A100345 (program): Triangle read by rows: T(n,k) = n*(n+k), 0<=k<=n.
- A100367 (program): Even numbers with two prime factors, not counting multiplicity.
- A100368 (program): Numbers of the form 2^k * p where k > 0 and p is an odd prime.
- A100371 (program): a(n) = 2^phi(n) - 1 = A066781(n) - 1.
- A100374 (program): Largest power of 2 dividing prime(n+1) - prime(n), the n-th consecutive prime difference.
- A100375 (program): a(n) is the n-th consecutive prime difference divided by the largest power of 2 which divides it.
- A100376 (program): a(n) is the largest number x such that for m=n to n+x-1, A006530(m) increases.
- A100381 (program): a(n) = 2^n*binomial(n,2).
- A100387 (program): a(n) is the largest number x such that for m=n to n+x-1, A006530(m) decreases.
- A100388 (program): a(n) = Bell(n) + Fibonacci(n).
- A100389 (program): a(n) = Bell(n) - Fibonacci(n).
- A100390 (program): Numbers n where A006530 has a local minimum.
- A100392 (program): Numbers k such that A006530(k-1) < A006530(k) > A006530(k+1).
- A100394 (program): a(n) is the subscript of the greatest prime factor of (2*prime(n) + 1).
- A100396 (program): Bell(n-1) + Fibonacci(n).
- A100397 (program): Bell(n-1) - Fibonacci(n).
- A100399 (program): a(n) = Fibonacci(n)^n.
- A100400 (program): Triangle read by rows: T(n,k) is the number of nonroot nodes of outdegree k (0<=k<=n-1) in all non-crossing trees with n edges.
- A100401 (program): Digital root of 3^n.
- A100402 (program): Digital root of 4^n.
- A100403 (program): Digital root of 6^n.
- A100404 (program): a(n) = L(n) * n! where L(n) are the Lucas numbers.
- A100412 (program): a(n) = 8*10^n - 7.
- A100413 (program): Numbers n such that n is reversal(n)-th even composite number (n is A004086(n)-th even composite number).
- A100428 (program): Bisection of Kolakoski sequence A000002.
- A100429 (program): Bisection of Kolakoski sequence A000002.
- A100430 (program): Bisection of A002417.
- A100431 (program): Bisection of A002417.
- A100432 (program): Bisection of A005349.
- A100433 (program): Bisection of A005349.
- A100434 (program): G.f.: (1+x)*(3+x)/(1+6*x^2+x^4).
- A100441 (program): a(n) is the denominator of f(n) where f(1) = 2 and f(n+1) is the solution of x + Sum_{i=1..n} f(i) = x * Product_{i=1..n} f(i).
- A100442 (program): Binomial transform of A003418.
- A100443 (program): Inverse binomial transform of A003418.
- A100444 (program): Bisection of A000255.
- A100445 (program): Bisection of A000255.
- A100446 (program): Bisection of A000031.
- A100447 (program): Bisection of A000031.
- A100448 (program): Number of triples (i,j,k) with 1 <= i <= j < k <= n and gcd{i,j,k} = 1.
- A100449 (program): Number of ordered pairs (i,j) with |i| + |j| <= n and gcd(i,j) <= 1.
- A100450 (program): Number of ordered triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.
- A100451 (program): a(n) = 0 for n <= 2; for n >= 3, a(n) = (n-2)*floor((n^2-2)/(n-2)).
- A100454 (program): a(n) = sum of n-th column in array in A100452.
- A100455 (program): a(n) = 2^n + sin(n*Pi/2).
- A100462 (program): Leading diagonal of array in A100461.
- A100463 (program): a(n) = 2^(n-1) - A100462(n).
- A100466 (program): Semiprimes of special form: sum of an integer k and the k-th semiprime.
- A100470 (program): n appears A055642(n) times (appearances equal number of decimal digits).
- A100472 (program): Inverse modulo 2 modulo transform of 9^n.
- A100477 (program): a(n) = 3*a(n-1)+2*a(n-2)+a(n-3) if n>=3 else a(n) = n.
- A100479 (program): Prime(2n-1) + prime(2n).
- A100481 (program): Greatest prime factor in A095117(n) = greatest prime factor in n + pi(n) where pi(n) is the prime counting function = greatest prime factor in n + A000720(n).
- A100484 (program): The primes doubled.
- A100486 (program): a(n) = pi(n) + n-th prime, where pi(n) = A000720(n) is the prime counting function.
- A100490 (program): Odd numbers ending in {1,3,7,9} that are not primes.
- A100493 (program): a(n) = n + n-th semiprime.
- A100494 (program): Primes of the form 47n+3.
- A100500 (program): a(n) = prime(3n-2) + prime(3n-1) + prime(3n).
- A100503 (program): Bisection of A000125.
- A100504 (program): a(n) = (4*n^3 + 6*n^2 + 8*n + 6)/3.
- A100510 (program): Bisection of A005425.
- A100511 (program): a(n) = Sum_{j=0..n} Sum_{k=0..n} binomial(n,j)*binomial(n,k)*max(j,k).
- A100512 (program): Numerator of Sum_{k=0..n} 1/C(2n,2k).
- A100513 (program): Denominator of Sum_{k=0..n} 1/C(2n,2k).
- A100516 (program): Numerator of Sum_{k=0..n} 1/binomial(n,k)^2.
- A100517 (program): Denominator of Sum_{k=0..n} 1/binomial(n,k)^2.
- A100518 (program): Numerator of Sum_{k=0..n} 1/binomial(n,k)^3.
- A100519 (program): Denominator of Sum_{k=0..n} 1/binomial(n,k)^3.
- A100520 (program): Numerator of Sum_{k=0..2*n} (-1)^k/binomial(2*n, k)^2.
- A100521 (program): Denominator of Sum_{k=0..2*n} (-1)^k/binomial(2*n, k)^2.
- A100525 (program): Bisection of A048654.
- A100526 (program): Number of local binary search trees (i.e., labeled binary trees such that every left child has a smaller label than its parent and every right child has a larger label than its parent) with n vertices such that the root has only one child.
- A100528 (program): a(0) = 1, a(n+1) = a(n)^2 + 1 - floor(log_2(a(n))).
- A100530 (program): Numbers == 0,2,5,9 modulo 10.
- A100531 (program): a(n) = a(n-1) + (2*n - 1) mod 8 + 1 with a(0)=1.
- A100532 (program): The first four numbers of this sequence are the primes 2,3,5,7. The other terms are calculated by adding the previous four terms.
- A100534 (program): Number of partitions of 2*n into parts of two kinds.
- A100535 (program): Number of partitions of 2*n + 1 into parts of two kinds.
- A100536 (program): a(n) = 3*n^2 - 2.
- A100537 (program): Triangle read by rows: T(n,k) is the number of Dyck n-paths whose first descent has length k.
- A100538 (program): Volume of the 3-dimensional box of sides of length equal to consecutive Padovan numbers (A000931). These boxes form a spiral in three dimensions similar to the spiral of Fibonacci boxes in two dimensions.
- A100542 (program): Two-color Rado numbers R(0,n).
- A100545 (program): Expansion of (7-2*x) / (1-3*x+x^2).
- A100546 (program): Difference between the smallest semiperimeter (see A063655) and its integer log (A001414) equals 1.
- A100549 (program): Let n = 2^e_2 * 3^e_ * 5^e_ * … be the prime factorization of n; then a(n) = largest prime <= 1 + max{e_2, e_3, e_5, …}; a(1) = 1 by convention.
- A100550 (program): If n>3 a(n)=a(n-1)+2*a(n-2)+3*a(n-3) else a(n)=n
- A100551 (program): Coefficient list of ChebyshevU(k,1-x).
- A100555 (program): Smallest square that is equal to the sum of n not-necessarily-distinct primes plus 1.
- A100560 (program): Numerator of Sum_{k=0..[n/2]} 1/binomial(n,k).
- A100561 (program): Denominator of Sum_{k=0..[n/2]} 1/binomial(n,k).
- A100565 (program): a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.
- A100566 (program): 2*a(n) = prime(prime(n)) + prime(prime(2n)).
- A100567 (program): Prime-indexed primes as n runs through the integers congruent to 0 or 1 mod 3.
- A100571 (program): Cubes m^3 such that m^3 is the sum of m-1 consecutive primes plus a larger prime.
- A100575 (program): Half the number of permutations of 0..n with exactly two maxima.
- A100577 (program): Number of sets of divisors of n with an odd sum.
- A100583 (program): Number of triangles in an n X n grid of squares with diagonals.
- A100585 (program): a(n+1) = a(n)+floor(a(n)/3), a(1) = 3.
- A100586 (program): Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off every 5th term. Repeat, always crossing off every 5th term of those that remain. The numbers that are left form the sequence.
- A100587 (program): Number of nonempty subsets of divisors of n.
- A100606 (program): a(n) = n^4 + n^3 + n.
- A100612 (program): a(n) = (0! + 1! + … + (p-1)!) mod p, where p = prime(n).
- A100613 (program): Number of elements in the set {(x,y): 1 <= x,y <= n, gcd(x,y) > 1}.
- A100617 (program): There are n people in a room. The first half (i.e., floor(n/2)) of them leave, then 1/3 (i.e., floor of 1/3) of those remaining leave, then 1/4, then 1/5, etc.; sequence gives number who remain at the end.
- A100618 (program): Initially there are n people in a room. At each step, if there are currently M people in the room, [M/k^2] of them leave, for k = 2, 3, … Sequence gives number who are left at the end.
- A100619 (program): Fixed point of the morphism 1 -> 12, 2 -> 31, 3 -> 1, starting from a(1) = 1.
- A100622 (program): Expansion of e.g.f. exp( (1+2*x-sqrt(1-4*x))/4).
- A100626 (program): Numbers of the form 2^(2p+1) where p is prime.
- A100627 (program): 3^(2p + 1) where p is prime.
- A100628 (program): a(n) = 2^(3*prime(n) + 1).
- A100629 (program): a(n) = 2^(5*prime(n) + 1).
- A100630 (program): Array read by antidiagonals: T(m,n) = Sum(1<=i<=m) [ i*(n-1+i)! ]
- A100634 (program): a(n) is the decimal equivalent of the binary number whose k-th least significant bit is 1 iff k is a prime number and k <= n.
- A100635 (program): Number of 2 X 2 matrices with elements in {1,2,…,n} such that LCMs of rows and columns are n.
- A100637 (program): Trisection of A000720.
- A100638 (program): Successive powers of the matrix A=[1,2;3,4] written by rows in groups of 4.
- A100656 (program): a(n)=1 if a hexagonal number is a prime, otherwise 0.
- A100659 (program): Floor of measure (in degrees) of the internal angles of a regular polygon with n sides.
- A100661 (program): Quet transform of A006519 (see A101387 for definition). Also, least k such that n+k has at most k ones in its binary representation.
- A100662 (program): Primes of the form n^3 + (n+1)^2.
- A100665 (program): a(n) = round(F(n)^(1/2)) where F(n) is the n-th Fibonacci number (A000045).
- A100670 (program): Number of two-card Baccarat hands of point n.
- A100671 (program): A Graham-Pollak-like sequence with multiplier 3 instead of 2.
- A100672 (program): Second least-significant bit in the binary expansion of the n-th prime.
- A100675 (program): a(1) = 1; for n >= 1, a(n+1) = Sum_{k=1..n} gcd(k, a(n)).
- A100679 (program): Floor of cube root of tetrahedral numbers.
- A100683 (program): a(n) = a(n-1) + a(n-2) + a(n-3); a(0) = -1, a(1) = 2, a(2) = 2.
- A100688 (program): a(n) = prime(n) * 3^prime(n) - 1.
- A100689 (program): a(n) = prime(n) * 4^prime(n) - 1.
- A100690 (program): a(n) = p * 5^p - 1 where p=prime(n).
- A100691 (program): Number of self-avoiding paths with n steps on a triangular lattice in the strip Z x {0,1}.
- A100698 (program): Primes of the form n^3 - n + 1.
- A100699 (program): Number of ways to partition n into two squarefree numbers that are not prime.
- A100700 (program): n-th Fibonacci number minus n-th prime number.
- A100701 (program): a(n) = a(n-1) + a(n-2) + a(n-1)*a(n-2) for n>=2; a(0)=2, a(1)=3.
- A100702 (program): Number of layers of dough separated by butter in successive foldings of croissant dough.
- A100703 (program): (T(n-1) + T(n-2)) + T(n-1)*T(n-2) where T(0)=3, T(1)=5 and n >= 2.
- A100705 (program): a(n) = n^3 + (n+1)^2.
- A100706 (program): Bisection of A002275.
- A100709 (program): Trajectory of 1001 under “3x+1” map.
- A100710 (program): Characterized by a(n) XOR (a(n) + 1) = a(n) - n.
- A100714 (program): Number of runs in binary expansion of A000040(n) (the n-th prime number) for n > 0.
- A100716 (program): Numbers k such that p^p divides k for some prime p.
- A100720 (program): Expansion of g.f.: (3+x+2*x^2-2*x^3)/((1-2*x)*(1+x^2)).
- A100722 (program): Prime numbers whose binary representations are split into exactly five runs.
- A100723 (program): Prime numbers whose binary representations are split into exactly seven runs.
- A100726 (program): Prime numbers whose binary representations are split into a maximum of 7 runs.
- A100727 (program): Continued fraction expansion of (1/2) [tan(1) + sec(1)].
- A100732 (program): a(n) = (3*n)!.
- A100733 (program): a(n) = (4*n)!.
- A100735 (program): Inverse modulo 2 binomial transform of 2^n.
- A100736 (program): Inverse modulo 2 binomial transform of 3^n.
- A100737 (program): Inverse modulo 2 binomial transform of 4^n.
- A100738 (program): Inverse modulo 2 binomial transform of 5^n.
- A100740 (program): Inverse modulo 2 binomial transform of 7^n.
- A100744 (program): Inverse modulo 2 binomial transform of (-2)^n.
- A100745 (program): Modulo 2 binomial transform of the Jacobsthal numbers J(n).
- A100746 (program): Inverse modulo 2 binomial transform of Jacobsthal numbers J(n).
- A100747 (program): A modular recurrence.
- A100749 (program): Triangle read by rows: T(n,k)=number of 231- and 312-avoiding permutations of [n] having k fixed points.
- A100752 (program): a(n) is the number of positive integers <= 10^n that are divisible by no prime exceeding 3.
- A100760 (program): Primes of the form 47n+5.
- A100762 (program): Let n = 2^e_2 * 3^e_3 * 5^e_5 * … be the prime factorization of n and let P(n) = A100549(n); then a(n) = Product_{ q <= P(n) } q^e_q; a(1) = 1 by convention.
- A100764 (program): a(1) = 1, a(2) = 2, a(3) = 3, a(n) = least number not the sum of three or fewer previous terms.
- A100768 (program): a(n) = p * (n^p) - 1 where p = prime(n).
- A100774 (program): a(n) = 2*(3^n - 1).
- A100775 (program): a(n) = 97*n + 101.
- A100776 (program): a(n) = 997 * n + 1009.
- A100777 (program): Square-factorial numbers: a(1) = 1, a(n+1) = a(n) * largest square divisor of (n+1).
- A100779 (program): Expansion of (1+t^2+4*t^3+2*t^4+t^5+3*t^6)/((1-t)^2*(1-t^2)*(1-t^3)^2).
- A100789 (program): First differences of A000543.
- A100790 (program): First differences of A047780.
- A100791 (program): Group the natural numbers so that the n-th group contains n(n+1)/2 = T(n) terms: (1), (2,3,4), (5,6,7,8,9,10), (11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35),… The n-th row of the following triangle is formed from the sum of first n terms, next n-1 terms,next n-2 terms … of the n-th group; e.g. third row is (5+6+7), (8+9), (10) or 18,17,10. Sequence contains the triangle read by rows.
- A100792 (program): Group the natural numbers so that the n-th group contains n(n+1)/2 = T(n) terms: (1), (2,3,4), (5,6,7,8,9,10), (11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35),… The r-th term of the n-th row of the following triangle is the sum of the next r terms of the n-th group, e.g. third row is (5),(6+7), (8+9+10) or 5,13,27. Sequence contains the triangle read by rows.
- A100795 (program): n occurs n times, as early as possible subject to the constraint that no two successive terms are identical.
- A100802 (program): a(n) = least k >= 0 such that (n+k)/2 is prime.
- A100803 (program): A100802(m) where A100802(m) > A100802(m-1).
- A100808 (program): Quet transform (see A101387 for definition) of Kolakoski sequence A000002.
- A100810 (program): a(n) = 0 if prime(n) + 2 = prime(n+1), otherwise 1.
- A100814 (program): Digits 9 to 0 are written in order with increasing number of digits for each member of the sequence. Leading zeros are counted, but are not written down.
- A100817 (program): Product of the digits of n, each doubled.
- A100818 (program): For a given unrestricted partition pi, let P(pi)=lambda(pi), if mu(pi)=0. If mu(pi)>0 then let P(pi)=nu(pi), where nu(pi) is the number of parts of pi greater than mu(pi), mu(pi) is the number of ones in pi and lambda(pi) is the largest part of pi.
- A100820 (program): Number of odd numbers between prime(n) and prime(n+1).
- A100821 (program): a(n) = 1 if prime(n) + 2 = prime(n+1), otherwise 0.
- A100822 (program): Triangle read by rows: T(n,k) is the number of deco polyominoes of height n with k cells in the first column. (A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column).
- A100824 (program): Number of partitions of n with at most one odd part.
- A100828 (program): Expansion of (1+2*x-2*x^3-3*x^2)/((x-1)*(x+1)*(x^2+2*x-1)).
- A100830 (program): Smallest number with same digital root as n but distinct from n and all earlier occurrences.
- A100832 (program): Amenable numbers: n such that there exists a multiset of integers (s(1), …, s(n)) whose size, sum and product are all n.
- A100833 (program): Smallest positive palindrome-free and squarefree sequence.
- A100836 (program): a(n) is the smallest value k > 1 such that k^2 - 1 is divisible by n^2.
- A100843 (program): F(P(n)) where P(n) is the unrestricted partition number of n and F(n) is the Fibonacci number.
- A100845 (program): a(n) = L(P(n)), where P = A000041 (partition numbers) and L = A000032 (Lucas numbers).
- A100851 (program): Triangle read by rows: T(n,k) = 2^n * 3^k, 0<=k<=n.
- A100852 (program): Triangle read by rows: T(n,k) = 2^k * 3^n, 0 <= k <= n.
- A100855 (program): n*(n^3-n^2+n+1)/2.
- A100856 (program): a(n) = (prime(n) - 1)! + prime(n).
- A100859 (program): Beginning with 3, increasing primes such that no two adjacent terms are congruent mod 4.
- A100860 (program): Lesser of two consecutive primes of forms 4k+3 and 4k+1 respectively.
- A100861 (program): Triangle of Bessel numbers read by rows: T(n,k) is the number of k-matchings of the complete graph K(n).
- A100862 (program): Triangle read by rows: T(n,k) is the number of k-matchings of the corona K’(n) of the complete graph K(n) and the complete graph K(1); in other words, K’(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v’ and the edge vv’.
- A100868 (program): a(n) = Sum_{k>0} k^(2n-1)/phi^(2k) where phi = (1+sqrt(5))/2 = A001622.
- A100872 (program): a(n) = (1/sqrt(5)) * Sum_{k>0} k^(2n)/phi^(2k) where phi = (1+sqrt(5))/2 = A001622.
- A100876 (program): Least number of squares that sum to prime(n).
- A100877 (program): Greater of two consecutive primes of form 4k+3 and 4k+1 respectively.
- A100879 (program): a(n) = n^sigma(n).
- A100886 (program): Expansion of x*(1+3*x+2*x^2)/((1+x+x^2)*(1-x-x^2)).
- A100887 (program): Expansion of (-1+2x+2x^2)/((1+x+x^2)(1-x-x^2)).
- A100888 (program): Expansion of (3+x-x^2)/((1+x+x^2)(1-x-x^2)).
- A100892 (program): a(n) = (2*n-1) XOR (2*n+1), bitwise.
- A100898 (program): Triangle read by rows: T(n,k) is the number of k-matchings of the fan graph on n+1 vertices (i.e., the join of the path graph on n vertices with one extra vertex).
- A100915 (program): Numbers n such that n plus n-th semiprime is semiprime.
- A100916 (program): Sum of a semiprime and its semiprime index is a new semiprime.
- A100921 (program): n appears A023416(n) times (appearances equal number of 0-bits).
- A100922 (program): n appears A000120(n) times (appearances equal number of 1-bits).
- A100923 (program): a(n) = 1 iff 6*n+1 and 6*n-1 are both prime numbers (0 otherwise).
- A100930 (program): Semiprimes of the form (2*p+1)*(p-1)/2, p prime.
- A100937 (program): Main diagonal of symmetric square array A100936.
- A100938 (program): Self-convolution of A092684.
- A100948 (program): Irregular triangle with T(n,1) = floor(n!/3), T(n,2) = n!/2, T(n,3) = n!, read by rows.
- A100954 (program): Decimal expansion of 7/2 - sqrt(2)/4.
- A100959 (program): Non-semiprimes.
- A100963 (program): a(1)=1. a(n) = a(n-1) + sum of the triangular numbers which are among the first (n-1) terms of the sequence.
- A100990 (program): a(n) = n^21 mod 100.
- A100992 (program): Indices k of Fibonacci numbers F(k) (A000045) that are divisible by k+1.
- A100993 (program): Indices k of Fibonacci numbers F(k) (A000045) that are divisible by k-1.
- A100994 (program): If n is a prime power p^m, m >= 1, then n, otherwise 1.
- A100995 (program): If n is a prime power p^m, m >= 1, then m, otherwise 0.
- A101000 (program): Periodic sequence with period 3.
- A101028 (program): Numerator of partial sums of a certain series. First member (m = 2) of a family.
- A101029 (program): Denominator of partial sums of a certain series.
- A101030 (program): Triangle read by rows: T(n,k) = number of functions from an n-element set into but not onto a k-element set.
- A101031 (program): Triangle read by rows: T(n,k) = (1/k) times the number of functions from an n-element set into but not onto a k-element set.
- A101035 (program): Dirichlet inverse of the gcd-sum function (A018804).
- A101037 (program): Triangle read by rows: T(n,1) = T(n,n) = n and for 1<k<n: T(n,k) = floor((T(n-1,k-1)+T(n-1,k))/2).
- A101038 (program): Inverse to sequence matrix for odd numbers.
- A101040 (program): If n has one or two prime-factors then 1 else 0.
- A101041 (program): Number of numbers not greater than n having no more than two prime factors.
- A101048 (program): Number of partitions of n into semiprimes (a(0) = 1 by convention).
- A101052 (program): Number of preferential arrangements of n labeled elements when only k <= 3 ranks are allowed.
- A101053 (program): a(n) = n! * Sum_{k=0..n} Bell(k)/k! (cf. A000110).
- A101054 (program): E.g.f.: exp(exp(x)-1)/(1-x)^2.
- A101055 (program): E.g.f.: exp(exp(x)-1)/(1-x)^3.
- A101080 (program): Table of Hamming distances between binary vectors representing i and j, for i >= 0, j >= 0, read by antidiagonals.
- A101082 (program): Numbers n such that binary representation contains bit strings “10” and “01” (possibly overlapping).
- A101084 (program): Numbers k such that 97*k + 101 is a prime.
- A101086 (program): Numbers k for which 997*k + 1009 is prime.
- A101089 (program): Second partial sums of fourth powers (A000583).
- A101090 (program): Third partial sums of fourth powers (A000583).
- A101091 (program): Fourth partial sums of fourth powers (A000583).
- A101092 (program): Second partial sums of fifth powers (A000584).
- A101093 (program): Second partial sums of sixth powers (A001014).
- A101094 (program): a(n) = n*(n+1)*(n+2)*(n+3)*(1+3*n+n^2)/120.
- A101095 (program): Fourth difference of fifth powers (A000584).
- A101096 (program): Third differences of fifth powers (A000584).
- A101097 (program): a(n) = n*(n+1)*(n+2)*(n+3)*(n+4)*(2 + 4*n + n^2)/840.
- A101098 (program): a(1)=1; thereafter, a(n+1) = 20*n^3 + 10*n.
- A101099 (program): Third partial sums of fifth powers (A000584).
- A101100 (program): The first summation of row 5 of Euler’s triangle - a row that will recursively accumulate to the power of 5.
- A101101 (program): a(1)=1, a(2)=5, and a(n)=6 for n>=3.
- A101102 (program): Fifth partial sums of cubes (A000578).
- A101103 (program): Partial sums of A101104. First differences of A005914.
- A101104 (program): a(1)=1, a(2)=12, a(3)=23, and a(n)=24 for n>=4.
- A101105 (program): Row sums of triangle A101224, which is related to the Flavius sieve (A000960).
- A101107 (program): Sorted and uniqued list of class numbers (number of conjugacy classes) of all non-Abelian simple groups.
- A101109 (program): Number of sets of lists (sequences) of n labeled elements with k=3 elements per list.
- A101119 (program): Nonzero differences of A006519 (highest power of 2 dividing n) and A003484 (Radon function).
- A101120 (program): Records in A101119, which forms the nonzero differences of A006519 and A003484.
- A101123 (program): Numbers n for which 7*n + 11 is prime.
- A101124 (program): Number triangle associated to Chebyshev polynomials of first kind.
- A101125 (program): Row sums of a Chebyshev number triangle.
- A101127 (program): McKay-Thompson series of class 12D for the Monster group.
- A101135 (program): a(1)=1. a(n) = a(n-1) + sum of the squares which are among the first (n-1) terms of the sequence.
- A101156 (program): a(n) = 2*Fibonacci(n) + 8*Fibonacci(n-5).
- A101157 (program): Let j be the smallest integer for which n+(n+1)+…+(n+j) is a square, say k^2; then a(n)=k.
- A101158 (program): Let j be the smallest integer for which n+(n+1)+…+(n+j) is a square; sequence gives the squares.
- A101159 (program): Let j be the smallest integer for which n+(n+1)+…+(n+j) is a square; then a(n) = n+j.
- A101160 (program): a(n) is the smallest integer j for which n+(n+1)+…+(n+j) is a square.
- A101161 (program): A number triangle associated with the Chebyshev polynomials of the first kind.
- A101162 (program): Row sums of a Chebyshev number triangle.
- A101164 (program): Triangle read by rows: Delannoy numbers minus binomial coefficients.
- A101165 (program): a(n) = (7*n^3 + 6*n^2 + 5*n) / 6.
- A101166 (program): a(n) = (15*n^4 + 22*n^3 + 45*n^2 + 14*n) / 24.
- A101168 (program): Lengths of successive words (starting with a) under the substitution: {a -> aab, b -> aac, c -> a}.
- A101169 (program): Lengths of successive words (starting with a) under the substitution: {a -> aab, b -> aac, c -> ab}.
- A101184 (program): a(n) = n + pi(n) + pi(pi(n)) + pi(pi(pi(n))) + pi(pi(pi(pi(n)))) + …
- A101195 (program): Expansion of psi(x^3) / psi(x) in powers of x where psi() is a Ramanujan theta function.
- A101197 (program): Lengths of successive words (starting with a) under the substitution: {a -> aab, b -> aac, c -> b}.
- A101201 (program): Maximal number of kings in the toroidal king’s graph on an n X n board such that each king is attacking no more than four other kings.
- A101202 (program): Multiples of 142857.
- A101203 (program): a(n) = sum of nonprimes <= n.
- A101207 (program): For each prime power n, a(n) is the number of positive integers that have n as their greatest prime power.
- A101213 (program): a(n) = n * (n+1)^2 * (n+2)^3.
- A101214 (program): a(n) = n * (n+1)^2 * (n+2)^3 * (n+3)^4.
- A101220 (program): a(n) = Sum_{k=0..n} Fibonacci(n-k)*n^k.
- A101230 (program): Number of partitions of 2n in which both odd parts and parts that are multiples of 3 occur with even multiplicities. There is no restriction on the other even parts.
- A101243 (program): Slowest increasing sequence where the absolute difference between the last digit of a(n) and the first digit of a(n+1) equals 9.
- A101256 (program): Sum of composites <= n.
- A101257 (program): Remainder when the least divisor of n greater than or equal to the square root of n (A033677(n)) is divided by the greatest divisor of n less than or equal to the square root of n (A033676(n)).
- A101263 (program): Decimal expansion of sqrt(2 - sqrt(3)), edge length of a regular dodecagon with circumradius 1.
- A101264 (program): a(n) = 1 if 2*n + 1 is prime, otherwise a(n) = 0.
- A101265 (program): a(1) = 1, a(2) = 2, a(3) = 6; a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3) for n>3.
- A101266 (program): First differences of A101402.
- A101269 (program): a(1)=0, a(2)=1 a(n+2)=(8*n^2+2*n+1)*a(n+1)-2*n*(2*n-1)^3*a(n).
- A101271 (program): Number of partitions of n into 3 distinct and relatively prime parts.
- A101272 (program): a(n)=n, n <=6; a(n)=6, n > 6.
- A101279 (program): a(1) = 1; a(2k) = a(k), a(2k+1) = k.
- A101289 (program): Inverse Moebius transform of 5-simplex numbers A000389.
- A101291 (program): Sum of all numbers with n digits.
- A101292 (program): a(n) = n! + Sum_{i=1..n} i.
- A101296 (program): n has the a(n)-th distinct prime signature.
- A101297 (program): Bisection of A001622 (decimal expansion of the golden ratio).
- A101299 (program): Numbers n such that -1 + Sum_{x=1..n} phi(x) is a prime number.
- A101300 (program): Second-smallest prime larger than n.
- A101301 (program): The sum of the first n primes, minus n.
- A101304 (program): a(n) = 2^(prime(n) + 1) + 1.
- A101305 (program): Begin with 0 and at each successive iteration append the next power of 10.
- A101306 (program): a(n) = Sum_{i=1..n} {last digit of prime(i)}.
- A101309 (program): Matrix logarithm of A047999 (Pascal’s triangle mod 2).
- A101310 (program): Sum((prime(k)*10^(k),k=1..n))).
- A101321 (program): Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.
- A101322 (program): a(n) = n - (least divisor of n greater than the square root of n) + (greatest divisor of n less than the square root of n) = n + A033676(n) - A033677(n).
- A101324 (program): Primes p such that p+1=C(q)=q-th composite and q is prime.
- A101328 (program): Recurring numbers in the count of consecutive composite numbers between balanced primes and their lower or upper prime neighbors.
- A101332 (program): a(n) = Knuth’s Fibonacci (or circle) square “n o n”.
- A101333 (program): A081254-A072762.
- A101334 (program): a(n) = n^n - (n+1)^(n-1).
- A101338 (program): Antidiagonal sums in A101321.
- A101339 (program): Prime(n)^prime(n)-prime(n).
- A101340 (program): a(n) = prime(n)^prime(n)+prime(n).
- A101342 (program): Fibonacci-Mersenne numbers.
- A101344 (program): Number of primes between prime(n) and 3prime(n).
- A101345 (program): a(n) = Knuth’s Fibonacci (or circle) product “2 o n”.
- A101346 (program): a(n) = binomial(2^n, n-1).
- A101348 (program): Arises from a particular cyclic transformation of the floretion - .5’i - .5i’ - .5’ij’ - .5’ik’ + .5’ji’ + .5’ki’.
- A101349 (program): Numbers of cubes between prime(n) and prime(n+1).
- A101351 (program): a(n) = 2^n-1 + Fibonacci(n).
- A101352 (program): Partial sums of A101351.
- A101353 (program): a(n) = Sum_{k=0..n} (2^k + Fibonacci(k)).
- A101356 (program): Binomial( 2*binomial(2*n,n-1),n-1).
- A101357 (program): Partial sums of A060354.
- A101361 (program): a(1) = a(2) = 1; for n > 2, a(n) = Knuth’s Fibonacci (or circle) product “a(n-1) o a(n-2)”.
- A101362 (program): a(n) = (n+1)*n^4.
- A101368 (program): The sequence solves the following problem: find all the pairs (i,j) such that i divides 1+j+j^2 and j divides 1+i+i^2. In fact, the pairs (a(n),a(n+1)), n>0, are all the solutions.
- A101369 (program): a(2n-1) = the smallest positive integer not occurring earlier in the sequence. a(2n) = the a(2n-1)th smallest positive integer among those not occurring earlier in the sequence.
- A101374 (program): a(n) = n*(n^3 - n + 2)/2.
- A101375 (program): a(n) = n*(n+1)*(n^2-2*n+2)/2.
- A101376 (program): a(n) = n^2*(n^3 - n^2 + n + 1)/2.
- A101377 (program): a(n) = n^2*(n^3-n+2)/2.
- A101378 (program): a(n) = n^2*(n^3+1)/2.
- A101381 (program): a(n) = n^2*(n+1)^2*(4*n^2 - 5*n + 4)/12.
- A101382 (program): a(n) = n*(n+1)*(2*n^3 - n^2 + 2)^2/6.
- A101383 (program): a(n) = n*(n+1)*(2*n^3 - n^2 + 2)/6.
- A101384 (program): a(n) = n*(n-1)^3*(n^2-n-1)/2.
- A101386 (program): Expansion of g.f.: (5 - 3*x)/(1 - 6*x + x^2).
- A101399 (program): a(0) = 1, a(1) = 2, a(2) = 5; for n >= 3, a(n) = a(n-1) + 2*a(n-2) + a(n-3).
- A101400 (program): a(n) = a(n-1) + 2*a(n-2) + a(n-3) - a(n-4).
- A101401 (program): Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the leftmost child of the root has degree k.
- A101402 (program): a(0)=0, a(1)=1; for n>=2, let k = smallest power of 2 that is >= n, then a(n) = a(k/2) + a(n-1-k/2).
- A101403 (program): Number of times that n occurs in A101402.
- A101404 (program): a(n) = n*A101403(n).
- A101405 (program): a(n) = n^(pi(n-1)).
- A101417 (program): Number of partitions of n into parts without powers of 2.
- A101418 (program): Floor of the area of a lens constructed using circular arcs of radius n.
- A101421 (program): Numbers which are the sum of two positive cubes and divisible by 7.
- A101423 (program): Number of different cuboids with volume p^3 * q^n, where p,q are distinct prime numbers.
- A101424 (program): Number of different cuboids with volume p^4 * q^n, where p,q are distinct prime numbers.
- A101425 (program): Number of different cuboids with volume p^5 X q^n, where p,q are distinct prime numbers.
- A101426 (program): Number of different cuboids with volume p^6 * q^n, where p,q are distinct prime numbers.
- A101427 (program): Number of different cuboids with volume (pq)^n, where p,q are distinct prime numbers.
- A101428 (program): Number of ways to write n as an ordered sum of a triangular number (A000217) and a square (A000290).
- A101432 (program): Each term is the number of letters in the Spanish name of the previous term.
- A101433 (program): Partial sums of A101402.
- A101435 (program): Dimension of a certain space of modular forms of weight 2 and level p^2, where p runs through the primes > 3 that are == 3 mod 4. See reference for precise definition.
- A101441 (program): n^prime(n+1).
- A101442 (program): a(n) = 9973*n + 10007.
- A101443 (program): Continued fraction expansion of (I_0(1/2)/I_1(1/2)-1)/2 = 1.56185896… (where I_n is the modified Bessel function of the first kind).
- A101444 (program): Numbers k such that (9973*k + 10007) is a prime.
- A101447 (program): Triangle read by rows: T(n,k) = (2*k+1)*(n+1-k), 0 <= k < n.
- A101448 (program): Nonnegative numbers k such that 2k + 11 is prime.
- A101455 (program): a(n) = 0 for even n, a(n) = (-1)^((n-1)/2) for odd n. Periodic sequence 1,0,-1,0,…
- A101457 (program): Prime digits in the decimal expansion of golden ratio phi (or tau) = (1 + sqrt 5 )/2.
- A101461 (program): Row maximum of Catalan triangle with zeros (A053121), i.e., maximum value of (m+1)*binomial(n+1,(n-m)/2)/(n+1) for given n with m same parity as n.
- A101463 (program): G.f.: (x^3+x^2+2*x+1)/(x^4+5*x^2+1).
- A101464 (program): Decimal expansion of sqrt(2-sqrt(2)), edge length of a regular octagon with circumradius 1.
- A101465 (program): Decimal expansion of 2-sqrt(2), square of the edge length of a regular octagon with circumradius 1.
- A101468 (program): Triangle read by rows: T(n,k)=(n+1-k)*(3*k+1).
- A101471 (program): Numbers n such that the number n11 is prime.
- A101472 (program): Numbers n such that the number n33 is prime.
- A101473 (program): Boustrophedon transform of the Jacobsthal numbers.
- A101474 (program): Boustrophedon transform of the signed Jacobsthal numbers.
- A101478 (program): G.f. satisfies A(x) = x*(1+A)^4/(1+A^2).
- A101485 (program): a(n) = (4n)! / ( 4^n * (2n)! ).
- A101488 (program): Number of naturally embedded binary trees with n nodes that have no label greater than 0.
- A101490 (program): G.f. satisfies A(x) = x*(1+A^2)^2/(1-A+A^2).
- A101492 (program): Triangle read by rows: T(n,k) = (n-k+1)*(4*k+1).
- A101493 (program): Triangle read by rows: T(n,k) = (n+1)*(2*(n+1)-1) - k*(2*k-1).
- A101495 (program): Column 1 of triangle A101494.
- A101496 (program): Expansion of (1-x^2)/(1-x-x^2+x^3+x^4).
- A101497 (program): Expansion of (1-x^2)/(1-2x+2x^3+x^4).
- A101498 (program): Expansion of (1-x^2)/(1-3x+x^2+3x^3+x^4).
- A101499 (program): A Chebyshev transform of the Catalan numbers.
- A101500 (program): A Chebyshev transform of the central binomial numbers.
- A101501 (program): Number of walks between adjacent nodes on C_5 tensor J_2.
- A101502 (program): Number of closed walks on C_5 tensor J_2.
- A101503 (program): Numbers n such that 11*n + 101 is prime.
- A101505 (program): Smallest m such that A101504(m) = n.
- A101508 (program): Product of binomial matrix and the Mobius matrix A051731.
- A101509 (program): Binomial transform of tau(n) (see A000005).
- A101514 (program): Shifts one place left under the square binomial transform (A008459): a(0) = 1, a(n) = Sum_{k=0..n-1} C(n-1,k)^2*a(k).
- A101550 (program): Lopsided (or biased) numbers: numbers n such that the largest prime factor of n is > 2*sqrt(n).
- A101551 (program): C(n-2,2)+C(n-5,5)+…+C(n-(3*floor((n-3)/3)+2),3*floor((n-3)/3)+2).
- A101552 (program): C(n-3,3)+C(n-7,7)+…+C(n-(4*floor((n-4)/4)+3),4*floor((n-4)/4)+3).
- A101553 (program): A modular recurrence.
- A101554 (program): Second inverse mod 2 binomial transform of 2^n.
- A101555 (program): Convolution of A010060 and A000244.
- A101556 (program): A Thue-Morse convolution.
- A101557 (program): Numbers k such that 101*k + 1009 is prime.
- A101561 (program): a(n)=(-1)^n*coefficient of x^n in sum{k>=1,x^(k-1)/(1+3x^k)}.
- A101562 (program): a(n)=(-1)^n*coefficient of x^n in sum{k>=1,x^(k-1)/(1+4x^k)}.
- A101563 (program): a(n)=(-1)^n*coefficient of x^n in sum{k>=1,x^(k-1)/(1+10x^k)}.
- A101566 (program): Binary partition sequence matrix.
- A101596 (program): G.f.: c(2*x)^4, where c(x) is the g.f. of A000108.
- A101600 (program): G.f.: c(3x)^2, c(x) the g.f. of A000108.
- A101601 (program): G.f.: c(3x)^3, c(x) the g.f. of A000108.
- A101602 (program): G.f.: c(3x)^4, c(x) the g.f. of A000108.
- A101603 (program): Riordan array (1/(1-x^2), x(1+x)/(1-x)).
- A101604 (program): a(n) = 2*a(n-1) + 5*a(n-2) + 2*a(n-3).
- A101605 (program): a(n) = 1 if n is a product of exactly 3 (not necessarily distinct) primes, otherwise 0.
- A101606 (program): a(n) = number of divisors of n that have exactly three (not necessarily distinct) prime factors.
- A101607 (program): a(2n) = 7 - a(n), a(2n+1) = (n-1 mod 3) + 1.
- A101608 (program): Solution to Tower of Hanoi puzzle encoded in pairs with the moves (1,2),(2,3),(3,1),(2,1),(3,2),(1,3). The disks are moved from peg 1 to 2. For a tower of k disks use the first 2^k-1 number pairs.
- A101609 (program): a(n) = n! * Sum_{k=1..floor(n/2)} 1/k.
- A101610 (program): n! * Sum[k=1..ceiling(n/2), 1/k].
- A101611 (program): a(n) = n! * Sum_{k=ceiling(n/2)..n} 1/k.
- A101613 (program): (2n)! * Sum[k=n..2n, 1/k].
- A101614 (program): Fixed point of morphism 0 -> 01, 1 -> 02, 2 -> 10.
- A101615 (program): Number of representations of n as a sum of the Jacobsthal numbers A078008 (2 is allowed twice as a part).
- A101616 (program): Partial sums of a Jacobsthal representation sequence.
- A101617 (program): The trinomial transform (A027907) gives powers of 3, while the trinomial transform of this sequence shift one place left gives powers of 5.
- A101621 (program): Initial decimal digit of n^11.
- A101622 (program): A Horadam-Jacobsthal sequence.
- A101623 (program): Modular binomial transform of 10^n.
- A101624 (program): Stern-Jacobsthal numbers.
- A101625 (program): A bisection of the Stern-Jacobsthal numbers.
- A101626 (program): Initial decimal digit of n^12.
- A101627 (program): Numerator of partial sums of a certain series.
- A101628 (program): Denominator of partial sums of a certain series.
- A101630 (program): Denominator of partial sums of a certain series.
- A101631 (program): Numerator of partial sums of a certain series.
- A101632 (program): Denominator of partial sums of a certain series.
- A101634 (program): Subtract 1, multiply by 1, subtract 2, multiply by 2, etc.
- A101635 (program): Increasing primes of alternating congruences modulo 6.
- A101637 (program): a(n) = 1 if n is a 4-almost prime, that is a product of exactly four (not necessarily distinct) primes, 0 otherwise.
- A101642 (program): a(n) = Knuth’s Fibonacci (or circle) product “3 o n”.
- A101643 (program): First row of array in A101385.
- A101645 (program): Third row of array in A101385.
- A101650 (program): A Thue-Morse-Stern sequence.
- A101651 (program): a(n)=Product{k=0..n, 1+0^A010060(k)}/2.
- A101652 (program): a(n)=Product{k=0..n, 1+2^A010060(k)}/2.
- A101653 (program): a(n)=Product{k=0..n, 1+3^A010060(k)}/2.
- A101654 (program): a(n)=Product{k=0..n, 1+4^A010060(k)}/2.
- A101655 (program): a(n)=Product{k=0..n, 1+9^A010060(k)}/2.
- A101659 (program): Fixed point of morphism 0 -> 01, 1 -> 02, 2 -> 11.
- A101660 (program): Fixed point of morphism 0 -> 01, 1 -> 02, 2 -> 12.
- A101661 (program): Fixed point of morphism 0 -> 01, 1 -> 02, 2 -> 20.
- A101662 (program): Fixed point of morphism 0 -> 01, 1 -> 02, 2 -> 21.
- A101663 (program): Fixed point of morphism 0 -> 01, 1 -> 02, 2 -> 22.
- A101664 (program): Fixed point of morphism 0 -> 01, 1 -> 12, 2 -> 00.
- A101665 (program): Fixed point of morphism 0 -> 01, 1 -> 12, 2 -> 02.
- A101666 (program): Fixed point of morphism 0 -> 01, 1 -> 12, 2 -> 10, starting with 0.
- A101668 (program): Fixed point of morphism 0 -> 01, 1 -> 20, 2 -> 00.
- A101669 (program): Fixed point of morphism 0 -> 01, 1 -> 20, 2 -> 01.
- A101670 (program): Fixed point of morphism 0 -> 01, 1 -> 20, 2 -> 02.
- A101671 (program): Fixed point of morphism 0 -> 01, 1 -> 20, 2 -> 10.
- A101672 (program): Fixed point of morphism 0 -> 01, 1 -> 20, 2 -> 11.
- A101673 (program): Fixed point of morphism 0 -> 01, 1 -> 20, 2 -> 20.
- A101674 (program): Fixed point of morphism 0 -> 01, 1 -> 20, 2 -> 21.
- A101675 (program): Expansion of (1 - x - x^2)/(1 + x^2 + x^4).
- A101676 (program): a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) with initial terms 1,0,-2,-1,0.
- A101677 (program): a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) - 2*a(n-4) + 2*a(n-5) - a(n-6).
- A101680 (program): A modular binomial transform of 10^n.
- A101682 (program): Expansion of 2 - exp(-1 + sqrt(1-4x)).
- A101683 (program): Write exp(sqrt(1+x)-1) = Sum c(n) x^n/n!; then a(n) = numerator of c(n).
- A101686 (program): a(n) = Product_{i=1..n} (i^2 + 1).
- A101687 (program): a(n) = Sum_{k=1..n} floor(binomial(n,k)/k).
- A101688 (program): Once 1, once 0, repeat, twice 1, twice 0, repeat, thrice 1, thrice 0… and so on.
- A101691 (program): A modular binomial sum sequence.
- A101692 (program): A modular binomial sum transform of 2^n.
- A101693 (program): A modular binomial sum transform of 2^n.
- A101695 (program): a(n) = n-th n-almost prime.
- A101705 (program): Numbers n such that n = 12*reversal(n).
- A101711 (program): Main diagonal of A101646.
- A101741 (program): 4th row of A101646.
- A101744 (program): Triangular numbers which are 10-almost primes.
- A101745 (program): Indices of triangular numbers which are 10-almost primes. Indices of A101744.
- A101776 (program): Smallest k such that k^2 is equal to the sum of n not-necessarily-distinct primes plus 1.
- A101780 (program): Primes of the form 100*n + 3.
- A101785 (program): G.f. satisfies: A(x) = 1 + x*A(x)/(1 - x^2*A(x)^2).
- A101786 (program): G.f. satisfies: A(x) = 1 + x*A(x)/(1 - 2*x^2*A(x)^2).
- A101787 (program): |S(n)| where S(n) = {i : 1 <= i <= n and 4n-1 and 8n-1 are primes}.
- A101788 (program): n - A101787.
- A101789 (program): 8n-1 such that 4n-1 and 8n-1 are primes.
- A101790 (program): Numbers n such that 4n-1, 8n-1 and 16n-1 are primes.
- A101791 (program): 4n-1 such that 4n-1, 8n-1 and 16n-1 are primes.
- A101792 (program): Primes of the form 8*k-1 such that 4*k-1 and 16*k-1 are primes.
- A101793 (program): 16k-1 such that 4k-1, 8k-1 and 16k-1 are primes.
- A101803 (program): Nearest integer to n*(phi-1), where phi is golden ratio 1.618033988749895… (A001622).
- A101808 (program): Number of primes between two consecutive even numbers.
- A101810 (program): Number of compositions (ordered partitions) of the n-th prime into n nonnegative integers.
- A101813 (program): Odd Niven (or Harshad) numbers: odd numbers that are divisible by the sum of their digits.
- A101814 (program): Even Niven (or Harshad) numbers: even numbers that are divisible by the sum of their digits.
- A101817 (program): Triangle read by rows: T(n,h) = number of functions f:{1,2,…,n}->{1,2,…,n} such that |Image(f)|=h; h=1,2,…,n, n=1,2,3,… . Essentially A090657, but without zeros.
- A101818 (program): Triangle read by rows: (1/n)*T(n,h), where T(n,h) is the array in A101817.
- A101822 (program): Expansion of 1/(1-x-2*x^2-3*x^3).
- A101825 (program): G.f.: x*(1+x)^2/(1-x^3).
- A101850 (program): A Catalan transform of Pell(n+1).
- A101851 (program): a(n) = Sum_{k=0..n} (-1)^(n-k)*k*Stirling2(n,k).
- A101853 (program): a(n) = n*(20 + 15*n + n^2)/6.
- A101854 (program): a(n) = n*(n+1)*(n^2 + 21*n + 50)/24.
- A101855 (program): a(n) = n*(n+1)*(n+2)*(n+4)*(n+23)/120.
- A101859 (program): a(n) = 11 + (23*n)/2 + n^2/2.
- A101860 (program): a(n) = (3+n)*(2 + 33*n + n^2)/6.
- A101861 (program): n*(n+5)*(50+45*n+n^2)/24.
- A101862 (program): a(n) = n*(n+1)*(n+7)*(122+57*n+n^2)/120.
- A101863 (program): Main diagonal of A101858.
- A101864 (program): Wythoff BB numbers.
- A101865 (program): Third row of A101858.
- A101867 (program): Main diagonal of A101866.
- A101868 (program): a(n) = n + 2*ceiling(phi n), where phi = (1 + sqrt(5))/2. Row 1 of A101866.
- A101869 (program): Row 2 of A101866.
- A101870 (program): Row 3 of A101866.
- A101871 (program): Number of Abelian groups of order 2n+1.
- A101873 (program): Number of Abelian groups of order 4n+1.
- A101874 (program): Number of Abelian groups of order 4n+3.
- A101875 (program): Number of Abelian groups of order 4n+2.
- A101878 (program): Expansion of -LambertW(LambertW(-x))/x.
- A101879 (program): a(0) = 1, a(1) = 1, a(2) = 2; for n > 2, a(n) = 5*a(n-1) - 5*a(n-2) + a(n-3).
- A101881 (program): Write two numbers, skip one, write two, skip two, write two, skip three … and so on.
- A101882 (program): Write three numbers, skip one, write three, skip two, write three, skip three… and so on.
- A101883 (program): Write four numbers, skip one, write four, skip two, write four, skip three… and so on.
- A101890 (program): Sum C(n,2k)F(k), k=0..floor(n/2).
- A101891 (program): Sum C(n,2k)F(k+1), k=0..floor(n/2).
- A101892 (program): Sum C(n,2k)J(k), k=0..floor(n/2).
- A101893 (program): a(n) = sum_{k=0..floor(n/2)} C(n,2k)*Pell(k).
- A101904 (program): Number of leg-hypotenuse twin Pythagorean triples < 10^n.
- A101907 (program): Numbers n-1 such that the arithmetic mean of the first n Fibonacci numbers (beginning with F(0)) is an integer.
- A101909 (program): Number of primes between 2n and 4n.
- A101917 (program): G.f. satisfies: A(x) = 1/(1 + x*A(x^7)) and also the continued fraction: 1 + x*A(x^8) = [1; 1/x, 1/x^7, 1/x^49, 1/x^343, …, 1/x^(7^(n-1)), …].
- A101921 (program): a(2n) = a(n) + 2n - 1, a(2n+1) = 4n.
- A101925 (program): a(n) = A005187(n) + 1.
- A101926 (program): a(n) = 2^A101925(n).
- A101927 (program): E.g.f. of sin(arcsinh(x)) (odd powers only).
- A101928 (program): E.g.f. cos(arcsinh(x)) = sin(arccosh(x)) (even powers only).
- A101942 (program): Sequence f[n,4], where f[n,b] is as defined below.
- A101943 (program): Sequence f[n, 5], where f[n, b] is described in A101942.
- A101945 (program): a(n) = 6*2^n - n - 5.
- A101946 (program): a(n) = 6*2^n - 3*n - 5.
- A101950 (program): Product of A049310 and A007318 as lower triangular matrices.
- A101979 (program): Antidiagonal sums of A101309, which is the matrix logarithm of A047999 (Pascal’s triangle mod 2).
- A101986 (program): Maximum sum of products of successive pairs in a permutation of order n+1.
- A101987 (program): Product of nonzero digits of n-th prime.
- A101990 (program): a(1) = a(2) = 1, a(3) = 9; for n > 3, a(n) = 3*a(n-1) - 3*a(n-2) + 9*a(n-3).
- A102000 (program): Sequence generated from a lattice packing matrix.
- A102001 (program): A weighted tribonacci, (1,2,4).
- A102002 (program): Weighted tribonacci (1,2,4), companion to A102001.
- A102005 (program): Fixed point of the morphism 1 -> 12, 2 -> 111.
- A102026 (program): Number of n-bit strings that contain no more than 4 zeros and no more than 2 leading and 2 trailing zeros.
- A102036 (program): Triangle, read by rows, where the terms are generated by the rule: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + T(n-3,k-1), with T(0,0)=1.
- A102037 (program): Triangle of BitAnd(BitNot(n), k).
- A102038 (program): a(n+1) = n*a(n) + a(n-1), a(1)=1, a(2)=2.
- A102039 (program): a(n) = a(n-1) + last digit of a(n-1), starting at 1.
- A102040 (program): a(n) = a(n-1) + last digit of a(n-1), starting at 3.
- A102041 (program): a(n) = a(n-1) + last digit of a(n-1), starting at 7.
- A102042 (program): a(n) = a(n-1) + last digit of a(n-1), starting at 9.
- A102047 (program): Decimal expansion of -1/4 + log(2)/2.
- A102048 (program): Exponent of A046021(n) (least inverse of Kempner function A002034) when written as a power of A006530(n) (largest prime dividing n), with a(1) = 1.
- A102052 (program): Column 1 of triangle A102051, which is the matrix inverse of triangle A101275 (number of Schroeder paths).
- A102055 (program): Column 1 of A102054, the matrix inverse of A060083 (Euler polynomials).
- A102058 (program): Expansion of e.g.f. sin(arctanh(x)), odd powers only.
- A102059 (program): Expansion of e.g.f. cos(arctanh(x)), even powers only.
- A102066 (program): Sum of the first n primes, mod 6.
- A102068 (program): a(n) = P(n)!, where P(n) is the largest prime factor of n (with a(1) = 1).
- A102069 (program): Analogous to the oblong (promic or heteromecic) sequence formed but with reversal digits of factors multiplied.
- A102071 (program): Pairwise sums of general ballot numbers (A002026).
- A102080 (program): Number of matchings in the C_n X P_2 (n-prism) graph.
- A102083 (program): a(n) = 8*n^2 + 4*n + 1.
- A102084 (program): a(1) = 0; for n>0, write 2n=p+q (p, q prime), p*q maximal; then a(n)=p*q (see A073046).
- A102091 (program): Number of perfect matchings in the C_{2n} X P_3 graph (C_{2n} is the cycle graph on 2n vertices and P_3 is the path graph on 3 vertices).
- A102092 (program): a(n)< a(n+1) and: each digit is the absolute difference of the previous two; each digit is the absolute difference of the next two; each digit is the absolute difference of its two neighbors.
- A102094 (program): a(n) = (2*n-1)*(2*n+1)^2.
- A102105 (program): a(n) = (19*5^n - 16*3^n + 1) / 4.
- A102108 (program): Numbers of partitions of 2n into n primes.
- A102110 (program): Iterations during which LQTL cellular automaton passes through the origin. Iteration 4 8 32 64 416 and 832 result in completed Barbour-Chapman squares (see A094867).
- A102111 (program): Iccanobirt numbers (1 of 15): a(n) = a(n-1) + a(n-2) + R(a(n-3)), where R is the digit reversal function A004086.
- A102112 (program): Iccanobirt numbers (2 of 15): a(n) = a(n-1) + R(a(n-2)) + a(n-3), where R is the digit reversal function A004086.
- A102113 (program): Iccanobirt numbers (3 of 15): a(n) = a(n-1) + R(a(n-2)) + R(a(n-3)), where R is the digit reversal function A004086.
- A102114 (program): Iccanobirt numbers (4 of 15): a(n) = R(a(n-1)) + a(n-2) + a(n-3), where R is the digit reversal function A004086.
- A102115 (program): Iccanobirt numbers (5 of 15): a(n) = R(a(n-1)) + a(n-2) + R(a(n-3)), where R is the digit reversal function A004086.
- A102116 (program): Iccanobirt numbers (6 of 15): a(n) = R(a(n-1)) + R(a(n-2)) + a(n-3), where R is the digit reversal function A004086.
- A102117 (program): Iccanobirt numbers (7 of 15): a(n) = R(a(n-1)) + R(a(n-2)) + R(a(n-3)), where R is the digit reversal function A004086.
- A102118 (program): Iccanobirt numbers (8 of 15): a(n) = R(a(n-1) + a(n-2) + a(n-3)), where R is the digit reversal function A004086.
- A102126 (program): Minimum number of pieces needed to dissect a square into n smaller squares (not necessarily of the same size).
- A102129 (program): Expansion of (1 - x)*(1 + 2*x) / ((1 + x)*(1 - 4*x - x^2)).
- A102130 (program): Primes of the form 8*n^2 + 4*n + 1.
- A102147 (program): Second Eulerian transform of 1, 2, 3, 4, 5, … (A000027).
- A102148 (program): Numbers k such that 101*k + 11 is prime.
- A102166 (program): Numbers n such that 2*n^2 + 11*n + 101 is prime.
- A102188 (program): a(n) = Sum_{m=0..n} (-1)^m * binomial(n,m)*(1*3*5*…*(4m-1)).
- A102190 (program): Irregular triangle read by rows: coefficients of cycle index polynomial for the cyclic group C_n, Z(C_n,x), multiplied by n.
- A102206 (program): a(0) = 3, a(1) = 8, a(n+2) = 4*a(n+1) - a(n) - 2.
- A102207 (program): a(n) = 5a(n-1) - 5a(n-2) + a(n-3) with a(0) = 4, a(1) = 17, a(2) = 65.
- A102208 (program): Decimal expansion of the volume of an icosahedron with unit edge length.
- A102209 (program): Decimal expansion of ratio of both the surface area and the volume of an icosahedron to a dodecahedron with the same inradius.
- A102214 (program): Expansion of (1 + 4*x + 4*x^2)/((1+x)*(1-x)^3).
- A102217 (program): 3-Suzanne numbers.
- A102219 (program): 3-Monica numbers.
- A102221 (program): Column 0 of triangular matrix A102220, which equals [2*I - A008459]^(-1).
- A102230 (program): Triangle, read by rows, where each column equals the convolution of A032349 with the prior column, starting with column 0 equal to A032349 shift right.
- A102232 (program): Number of preferential arrangements of n labeled elements when at least k=three ranks are required.
- A102233 (program): Number of preferential arrangements of n labeled elements when at least k=3 elements per rank are required.
- A102237 (program): Smallest number equal to the product of n primes which is also equal to the sum of n distinct primes.
- A102239 (program): a(n) = (Sum_{i=0..n} 5^i) + 1 - (Sum_{i=0..n} 5^i) mod 2.
- A102244 (program): a(n) = exp(-1) * (n+1)! * Sum_{i>j>=0} j^n/i!.
- A102248 (program): Numbers n such that n111 is prime.
- A102249 (program): Numbers k such that k1111 is prime.
- A102250 (program): Indices of semiprime Haüy rhombic dodecahedral numbers.
- A102261 (program): a(n) = A002144(n) - A002145(n).
- A102269 (program): Primes p such that Q(sqrt(-21p)) has genus characters chi_{-3} = +1, chi_{-7} = +1.
- A102276 (program): a(n) = (a(n-1) * a(n-5) + a(n-3)^2) / a(n-6) with a(0) = … = a(5) = 1, a(n) = a(5-n) for all n in Z.
- A102283 (program): Period 3: repeat [0, 1, -1].
- A102285 (program): G.f. (1-x)/(7*x^2-6*x+1).
- A102286 (program): Total number of odd blocks in all partitions of n-set.
- A102287 (program): Total number of even blocks in all partitions of n-set.
- A102289 (program): Total number of odd lists in all sets of lists, cf. A000262.
- A102290 (program): Total number of even lists in all sets of lists, cf. A000262.
- A102291 (program): Total number of prime parts in all compositions of n.
- A102294 (program): Number of prime divisors (with multiplicity) of icosahedral numbers.
- A102296 (program): a(n) = (1/6)*(n+1)*(10*n^2 + 17*n + 12).
- A102297 (program): Number of distinct divisors of n+1 where n and n+1 are composite or twin composite numbers.
- A102298 (program): Number of prime divisors with multiplicity of n+1 where n and n+1 are composite or twin composite numbers.
- A102299 (program): Number of prime divisors with multiplicity of n where n and n+1 are composite or twin composite numbers.
- A102300 (program): Number of distinct prime divisors of n where n and n+1 are composite or twin composite numbers.
- A102301 (program): a(n) = ((3*n + 1)*2^(n+3) + 9 + (-1)^n)/18.
- A102302 (program): Largest number < n/2 coprime to n.
- A102303 (program): a(n) = (1/6) * (7^(n+1) - 3*(-1)^n + 2).
- A102304 (program): Sum of factors of numbers having exactly three prime factors.
- A102305 (program): a(n) = n^2 + 2*n + 3.
- A102307 (program): a(n) = Fibonacci(2n+1) * binomial(2n,n).
- A102309 (program): a(n) = Sum_{d divides n} moebius(d) * C(n/d,2).
- A102310 (program): Square array read by antidiagonals: Fibonacci(k*n).
- A102311 (program): Sum_{k=1..n} Fibonacci(k*(n-k)).
- A102312 (program): a(n) = Fibonacci(5*n).
- A102318 (program): A mean binomial transform of the Catalan numbers.
- A102319 (program): A mean binomial transform of the central binomial numbers.
- A102338 (program): Numbers n such that 10n+3 is prime.
- A102339 (program): Numbers k such that k*10^3 + 333 is prime.
- A102341 (program): Areas of ‘close-to-equilateral’ integer triangles.
- A102342 (program): Numbers k such that 10k + 7 is prime.
- A102343 (program): Numbers k such that k*10^3 + 777 is prime.
- A102344 (program): Numbers n such that the Diophantine equation (x+2)^3-x^3=2*n^2 has solutions.
- A102345 (program): a(n) = 3^n + (-1)^n.
- A102348 (program): Decimated primes: every 10th prime has been omitted.
- A102350 (program): Prime(144*n).
- A102352 (program): Numbers n such that n^3 can be partitioned into n primes such that n-1 are consecutive primes and the remaining prime is larger than the sum of the n-1 consecutive primes.
- A102354 (program): a(n) is the number of ways n can be written as k^2 * j, 0 < j <= k.
- A102362 (program): This finite sequence describes itself completely: there is 1 “0” in it, 1 “3”, 1 “4”, …, 2 “2” and 11 “1”.
- A102363 (program): Triangle read by rows, constructed by a Pascal-like rule with left edge = 2^k, right edge = 2^(k+1)-1 (k >= 0).
- A102364 (program): Number of terms in Fibonacci sequence less than n not used in Zeckendorf representation of n (the Zeckendorf representation of n is a sum of non-consecutive distinct Fibonacci numbers).
- A102366 (program): Number of subsets of {1,2,…,n} in which exactly half of the elements are less than or equal to sqrt(n).
- A102368 (program): Smallest k>0 such that n^k + 1 is not prime.
- A102370 (program): “Sloping binary numbers”: write numbers in binary under each other (right-justified), read diagonals in upward direction, convert to decimal.
- A102371 (program): Numbers missing from A102370.
- A102376 (program): a(n) = 4^A000120(n).
- A102377 (program): Gould’s sequence A001316 in binary.
- A102378 (program): a(n) = a(n-1) + a([n/2]) + 1, a(1) = 1.
- A102379 (program): a(n) is the minimal number of nodes in a binary tree of height n.
- A102386 (program): Numbers k such that k99999 is prime.
- A102389 (program): An evil count.
- A102390 (program): An odious count.
- A102391 (program): Evil numbers in evil places.
- A102392 (program): Odious numbers in odious places.
- A102393 (program): A wicked evil sequence.
- A102394 (program): A wicked odious sequence.
- A102395 (program): A mod 2 related Jacobsthal sequence.
- A102396 (program): A mod 2 related Jacobsthal sequence.
- A102397 (program): a(n) = concatenation of first n elements of Thue-Morse sequence A010059.
- A102403 (program): Number of Dyck paths of semilength n having no ascents of length 2.
- A102414 (program): Smallest semiprime greater than n-th prime.
- A102421 (program): To get a(n), start with 2n+1, multiply by 3 and add 1 and divide out any power of 2; then multiply by 3 and subtract 1 and divide out any power of 2.
- A102426 (program): Triangle read by rows giving coefficients of polynomials defined by F(0,x)=0, F(1,x)=1, F(n,x) = F(n-1,x) + x*F(n-2,x).
- A102427 (program): Triangle based on downward diagonals of A102426.
- A102428 (program): Central column of triangle A102427.
- A102429 (program): Row sums of A102427.
- A102436 (program): Number of matchings of the corona L’(n) of the ladder graph L(n)=P_2 X P_n. and the complete graph K(1); in other words, L’(n) is the graph constructed from L(n) by adding for each vertex v a new vertex v’ and the edge vv’.
- A102438 (program): a(n) = 100*n + 44.
- A102439 (program): a(n) = 100*n + 4.
- A102446 (program): a(n) = a(n-1) + 4*a(n-2) for n>1, a(0) = a(1) = 2.
- A102460 (program): a(n) = 1 if n is a Lucas number, else a(n) = 0.
- A102466 (program): Numbers such that the number of divisors is the sum of numbers of prime factors with and without repetitions.
- A102467 (program): Positive integers k, such that d(k) <> Omega(k) + omega(k), where d = A000005, Omega = A001222 and omega = A001221.
- A102470 (program): Numbers n such that denominator of Sum_{k=0 to n} 1/k! is n!.
- A102471 (program): Numbers n such that the denominator of Sum_{k=0 to 2n} 1/k! is (2n)!.
- A102472 (program): Triangle read by rows. Let S(k) be the sequence defined by F(0)=0, F(1)=1, F(n-1) + (n+k)*F(n) = F(n+1). E.g. S(0) = 0, 1, 1, 3, 10, 43, 225, 1393, 9976, 81201, … Then S(0), S(1), S(2), … are written vertically, next to each other, with the initial term of each on the next row down.
- A102473 (program): Triangle read by rows. Let S(k) be the sequence defined by F(0)=0, F(1)=1, F(n-1) + (n+k)*F(n) = F(n+1). E.g. S(0) = 0,1,1,3,10,43,225,1393,9976,81201, … Then S(0), S(1), S(2), … are written next to each other, vertically, with the initial term of each on the next row down. The order of the terms in the rows are then reversed.
- A102476 (program): Least modulus with 2^n square roots of 1.
- A102479 (program): Triangle read by rows: row n contains the numbers C(n,k)^(k-1) for 0 <= k <= n-1, n >= 1.
- A102480 (program): Triangle read by rows: row n contains the numbers C(n,k)^(k-1) for 0 <= k <= n, n >= 0.
- A102485 (program): a(n) = 5*3^n - 4*2^n.
- A102486 (program): a(n) = 4*a(n-1) - 5*a(n-2).
- A102487 (program): Numbers in base-12 representation that can be written with decimal digits.
- A102488 (program): Numbers in base-12 representation that cannot be written with decimal digits.
- A102489 (program): Take the decimal representation of n and read it as if it were written in hexadecimal.
- A102490 (program): Numbers in base-16 representation that cannot be written with decimal digits.
- A102508 (program): Suppose there are equally spaced chairs around a round table. Then a(n) is the maximal number of chairs for which there exists a seating arrangement of n people around the table such that if a waiter puts two glasses (randomly) on the table in front of two (different) chairs, it is always possible to turn the table so that the two glasses end up in front of two seated persons.
- A102511 (program): Sum(A008683(A102510(k)): k<=n).
- A102515 (program): a(n) = floor(1 + sqrt(2n + 1)).
- A102516 (program): Sum C(n-3k,3k+1), k=0..floor(n/6).
- A102517 (program): Expansion of (1+x^2)/((1-x+x^2)(1+2x^2)).
- A102518 (program): a(n) = Sum_{k=0..n} binomial(n, k) * Sum_{j=0..k} binomial(3k, 3j).
- A102519 (program): Decimal expansion of 1-(3*sqrt(3))/(4*Pi).
- A102520 (program): Decimal expansion of 1-(9*sqrt(3))/(8*Pi).
- A102526 (program): Antidiagonal sums of Losanitsch’s triangle (A034851).
- A102528 (program): a(n)=least positive integer not a(k) or a(k)+floor(k/2) for any k<n.
- A102529 (program): Complement of A102528.
- A102537 (program): Triangle T(n,k) read by rows: (1/n) * C(2n+k,k-1) * C(n,k).
- A102545 (program): Numbers k such that k999999 is prime.
- A102546 (program): Numbers t such that t1 is prime and t is a multiple of 10.
- A102547 (program): Triangle read by rows, formed from antidiagonals of the antidiagonals (A011973) of Pascal’s triangle (A007318).
- A102548 (program): Number of positive integers <= n that are expressible in the form u^2+v^2, with u and v integers.
- A102551 (program): a(n) = [prime(n)/(prime(n+1)-prime(n))], where [x] means the integer part of x.
- A102552 (program): a(n) = prime(n)-(prime(n+1)+prime(n-1))/2.
- A102554 (program): Numbers n such that p <> (n AND p) for at least one prime-factor p.
- A102556 (program): Numerator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle.
- A102557 (program): Denominator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle.
- A102558 (program): Numerator of the probability that (2n+1)-dimensional Gaussian random triangle has an obtuse angle.
- A102559 (program): Denominator of the probability that (2n+1)-dimensional Gaussian random triangle has an obtuse angle.
- A102560 (program): Expansion of (1-x^3)/(1-x^4).
- A102561 (program): a(n) = 2^floor(n/2)*((-1)^floor(n/2) + (-1)^n)/2.
- A102563 (program): A000120(A001045(n))-A001045(A000120(n)).
- A102564 (program): A000120(A078008(n))-A078008(A000120(n)).
- A102565 (program): a(n) = A102563(n) - A102564(n).
- A102566 (program): a(n) = {minimal k such that f^k(prime(n)) = 1} where f(m) = (m+1)/2^r, 2^r is the highest power of two dividing m+1.
- A102572 (program): a(n) = floor(log_4(n)).
- A102574 (program): a(n) is the sum of the distinct norms of the divisors of n over the Gaussian integers.
- A102584 (program): a(n) = 1/2 times the cancellation factor in reducing Sum_{k=0 to 2n+1} 1/k! to lowest terms.
- A102587 (program): T(n, k) = (-1)^n*2*[x^k] ChebyshevT(n, (1 - x)/2) with T(0,0) = 1, for 0 <= k <= n, triangle read by rows.
- A102590 (program): Inverse Boustrophedon transform of 2^n.
- A102591 (program): a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*3^(n-k).
- A102592 (program): a(n) = Sum_{k=0..n} binomial(2n+1, 2k)*5^(n-k).
- A102594 (program): Number of noncrossing trees with n edges in which no border edges emanate from the root.
- A102603 (program): 24n + 21.
- A102611 (program): Numbers n such that the number n77 is prime.
- A102613 (program): Numerator of the reduced fractions of the ratios of the number of primes less than n over the number of composites less than n.
- A102614 (program): Denominators of the reduced fractions of the ratios of the number of primes less than n over the number of composites less than n.
- A102615 (program): Nonprime numbers of order 2.
- A102616 (program): Nonprime numbers of order 3.
- A102617 (program): Primes p(n) such that n is a second-order nonprime number.
- A102619 (program): Numbers which are the sum of two positive cubes and divisible by 19.
- A102620 (program): Number of legal Go positions on a 1 X n board (for which 3^n is a trivial upper bound).
- A102622 (program): Nonprime numbers n concatenated n times.
- A102624 (program): Numbers k such that the number k23 is prime.
- A102625 (program): Triangle read by rows: T(n,k) is the sum of the weights of all vertices labeled k at depth n in the Catalan tree (1 <= k <= n+1, n >= 0).
- A102631 (program): a(n) = n^2 / (squarefree kernel of n).
- A102649 (program): Numbers n such that 11*n^2 + 11*n + 3 is prime.
- A102650 (program): a(n) = 4 * floor(28*2^n/15).
- A102651 (program): a(n) = 4 * floor(23*2^n/15).
- A102652 (program): a(n) = 4 * floor(24*2^n/15) = 4*A077854(n).
- A102653 (program): a(n) = 4 * floor(27*2^n/15).
- A102655 (program): Numbers that are the arithmetic mean of four successive primes.
- A102656 (program): Numbers n such that 11*n + 1 is prime.
- A102657 (program): Numbers n such that 11n^2 + 11n + 1 is prime.
- A102662 (program): Triangle read by rows: T(1,1)=1,T(2,1)=1,T(2,2)=3, T(k-1,r-1)+T(k-1,r)+T(k-2,r-1).
- A102669 (program): Number of digits >= 2 in decimal representation of n.
- A102670 (program): Number of digits >= 2 in the decimal representations of all integers from 0 to n.
- A102671 (program): Number of digits >= 3 in decimal representation of n.
- A102672 (program): Number of digits >= 3 in the decimal representations of all integers from 0 to n.
- A102673 (program): Number of digits >= 4 in decimal representation of n.
- A102674 (program): Number of digits >= 4 in the decimal representations of all integers from 0 to n.
- A102675 (program): Number of digits >= 5 in decimal representation of n.
- A102676 (program): Number of digits >= 5 in the decimal representations of all integers from 0 to n.
- A102677 (program): Number of digits >= 6 in decimal representation of n.
- A102678 (program): Number of digits >= 6 in the decimal representations of all integers from 0 to n.
- A102679 (program): Number of digits >= 7 in decimal representation of n.
- A102680 (program): Number of digits >= 7 in the decimal representations of all integers from 0 to n.
- A102681 (program): Number of digits >= 8 in decimal representation of n.
- A102682 (program): Number of digits >= 8 in the decimal representations of all integers from 0 to n.
- A102683 (program): Number of digits 9 in decimal representation of n.
- A102684 (program): Number of times the digit 9 appears in the decimal representations of all integers from 0 to n.
- A102685 (program): Partial sums of A055640.
- A102686 (program): Numbers k such that 11*k + 3 is prime.
- A102688 (program): a(n) = (1/n)*Sum_{k=1..n} k*2^gcd(n,k).
- A102689 (program): a(n) = 10000*n + 2468.
- A102690 (program): Number of n-expodigital numbers (i.e., numbers m such that m^n has exactly n decimal digits).
- A102691 (program): Least n-expodigital number (i.e., numbers m such that m^n has exactly n decimal digits).
- A102693 (program): a(n) is the number of digraphs (not allowing loops) with vertices 1,2,…,n that have a unique Eulerian tour (up to cyclic shift).
- A102699 (program): Number of strings of length n, using as symbols numbers from the set {1, 2, …, n}, in which consecutive symbols differ by exactly 1.
- A102700 (program): Numbers k such that 10*k + 9 is prime.
- A102701 (program): Non-“Ding!Bong!” numbers: positive numbers which are not a positive linear combination of 5’s and 7’s.
- A102702 (program): Expansion of (2-x-2*x^2-x^3)/(1-x-x^2)^2.
- A102703 (program): Numbers k such that 100*k+99 is prime.
- A102704 (program): Numbers k such that k999 is prime.
- A102705 (program): Numbers not of the form 7x + 9y with nonnegative x and y.
- A102710 (program): a(1) = 2, a(2) = 3, a(n+2) = a(n)*(a(n)+a(n+1)) - a(n+1).
- A102711 (program): Numbers k such that 11*k + 7 is prime.
- A102713 (program): Total sum of odd parts in all compositions of n.
- A102714 (program): Expansion of (x+2) / ((x+1)*(x^2-3*x+1)).
- A102715 (program): Triangle read by rows: T(n,k) is phi(binomial(n,k)), where phi is Euler’s totient function (0 <= k <= n).
- A102716 (program): Triangle read by rows: T(n,k) = sigma(binomial(n,k)) (0 <= k <= n), where sigma(m) is the sum of the positive divisors of m.
- A102721 (program): Numbers n such that 11*n + 13 is prime.
- A102724 (program): Sum of the first n pairs of consecutive primes (for example, a(3) = (2+3) + (3+5) + (5+7) = 25).
- A102728 (program): Array read by antidiagonals: T(n, k) = ((n+1)^k-(n-1)^k)/2.
- A102731 (program): Numbers k such that 11*k + 23 is prime.
- A102732 (program): Primes of the form 13n+5.
- A102733 (program): Numbers n such that 2*n + 101 is prime.
- A102734 (program): Primes of the form 23n+5.
- A102736 (program): Number of permutations of n elements without cycles whose length is a multiple of 3.
- A102741 (program): a(n) = 3^4 * binomial(n+3, 4).
- A102743 (program): Expansion of LambertW(-x)/(x*(x-1)).
- A102745 (program): Number of distinct prime factors of four consecutively concatenated primes.
- A102750 (program): Numbers n such that square of largest prime dividing n does not divide n.
- A102752 (program): Array read by antidiagonals: T(n, k) = ((n+2)^k-(n-1)^k)/3.
- A102753 (program): Decimal expansion of (Pi^2)/2.
- A102754 (program): Decimal expansion of (Pi^2)/2 - 4.
- A102756 (program): Triangle T(n,k), 0<=k<=n, read by rows defined by: T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.
- A102757 (program): a(n) = Sum_{i=0..n} C(n,i)^2 * i! * 3^i.
- A102761 (program): Same as A000179, except that a(0) = 2.
- A102762 (program): Curvatures of (largest) kissing circles along the circumference, starting with curvature = -1 and 2.
- A102765 (program): Array read by antidiagonals: T(n, k) = ((n+4)^k-(n-1)^k)/5.
- A102768 (program): Numbers k such that 23*k + 11 is prime.
- A102769 (program): Decimal expansion of the volume of a dodecahedron with each edge of unit length.
- A102770 (program): (p*q - 1)/2 where p and q are consecutive odd primes.
- A102771 (program): Decimal expansion of area of a regular pentagon with unit edge length.
- A102773 (program): a(n) = Sum_{i=0..n} binomial(n,i)^2*i!*4^i.
- A102781 (program): Number of positive even numbers less than the n-th prime.
- A102785 (program): G.f.: (x-1)/(-2*x^2 + 3*x^3 + 2*x - 1).
- A102786 (program): Integer part of n#/((p-3)# 3#), where p=preceding prime to n.
- A102790 (program): Integer part of n#/(p-3)#, where p=preceding prime to n.
- A102807 (program): a(n) is the square of one plus the number consisting of n 3’s.
- A102815 (program): “False so far” sequence.
- A102820 (program): Number of primes between 2*prime(n) and 2*prime(n+1), where prime(n) is the n-th prime.
- A102821 (program): Numbers n for which the square excess of n-th prime is prime.
- A102822 (program): a(n+1) is the integer part of sqrt(2*a(n)^2).
- A102827 (program): “True already”, base 10, start 1: a(n) is the least integer such that the sequence up to a(n-1) written in base 10 contains floor(a(n)/10) copies of the digit a(n) % 10, with a(0) = 1.
- A102828 (program): “True already”, base 2, start 1: a(n) is the least integer such that the sequence up to a(n-1) written in base 2 contains floor(a(n)/2) copies of the digit a(n) % 2, with a(0) = 1.
- A102831 (program): Number of n-digit 4th powers.
- A102839 (program): a(0) = 0, a(1) = 1, and a(n) = ((2*n - 1)*a(n-1) + 3*n*a(n-2))/(n - 1) for n >= 2.
- A102840 (program): a(0)=0, a(1)=1, a(n)=((2*n-1)*a(n-1)-5*n*a(n-2))/(n-1).
- A102841 (program): a(n) = ((9*n^2 + 33*n + 26)*2^n + (-1)^n)/27.
- A102845 (program): Number of prime factors of the sum of the first n odd primes.
- A102846 (program): a(0)=1, a(1)=1, a(n) = a(n-1)*a(n-2) + 2.
- A102851 (program): Primes of the form 19n + 5.
- A102852 (program): Primes whose squares are congruent to 5 (modulo 19).
- A102853 (program): Number of prime factors (with multiplicity) of number of points on surface of square pyramid.
- A102860 (program): Number of ways to change three non-identical letters in the word aabbccdd…, where there are n types of letters.
- A102861 (program): Numbers which in base 5 have digit-sum 4.
- A102862 (program): Numbers of prime factors of the sum of the first n primes.
- A102863 (program): a(n)=1 if at least one of the first n primes is a divisor of the sum of the first n primes; otherwise a(n)=0.
- A102865 (program): Base-4 digits are, in order, the first n terms of the sequence (1, 3, 21, 203, 2021, 20203, 202021, 2020203, 20202021, 202020203, … ).
- A102866 (program): Number of finite languages over a binary alphabet (set of binary words of total length n).
- A102871 (program): a(n) = a(n-3) - 5*a(n-2) + 5*a(n-1), a(0) = 1, a(1) = 3, a(2) = 10.
- A102875 (program): Let f(n) = n+2 if n == 1 mod 3, = n if n == 2 mod 3, = n-2 if n == 0 mod 3; then a(n) = Fibonacci(f(n)).
- A102877 (program): a(0) = 1, a(1) = 1; for n>0, a(2*n) = 3*a(2n-1), a(2*n+1) = 3*a(2*n) - 2*a(n-1).
- A102879 (program): A Chebyshev transform of the first kind of the central binomial numbers.
- A102880 (program): A Chebyshev transform of the first kind of the Catalan numbers.
- A102881 (program): Expansion of (1+x)/sqrt(1-4x^2-8x^3-4x^4).
- A102882 (program): Expansion of (1+2x)/sqrt((1-3x^2)(1+4x+5x^2)).
- A102885 (program): Index of n in the primes A000040 or nonprimes A141468.
- A102890 (program): A102889(n) is at least a(n).
- A102893 (program): Number of noncrossing trees with n edges and having degree of the root at least 2.
- A102898 (program): A Catalan-related transform of 3^n.
- A102899 (program): a(n) = ceiling(n/3)^2 - floor(n/3)^2.
- A102900 (program): a(n) = 3*a(n-1) + 4*a(n-2), a(0)=a(1)=1.
- A102901 (program): a(n) = a(n-1) + 6a(n-2), a(0)=1, a(1)=0.
- A102902 (program): a(n) = 9a(n-1) - 16a(n-2).
- A102905 (program): A modulo three sequential permutation the Fibonacci sequence (outer): permutation after the Fibonacci.
- A102909 (program): a(n) = Sum_{j=0..8} n^j.
- A102915 (program): Numbers n such that n3 is prime and n is a multiple of 10.
- A102928 (program): Reduced numerators of the harmonic means of the first n positive integers.
- A103115 (program): a(n) = 6*n*(n-1)-1.
- A103116 (program): a(n) = Sum_{i=1..n} (n-i+1)*phi(i).
- A103118 (program): Numbers k such that 100*k + 57 is prime.
- A103119 (program): Numbers n such that n0999 is prime.
- A103122 (program): Define a 1-1 correspondence between the integers Z and the nonnegative integers N by f(n) = A102370(n) if n >= 0, f(n) = A102371(-n) if n < 0; sequence gives a(n) = f^(-1)(n) for n >= 0.
- A103127 (program): Numbers congruent to {-1, 1, 3, 5} mod 16.
- A103128 (program): a(n) = floor(sqrt(2n-1)).
- A103131 (program): The product of the residues in [1,n] relatively prime to n, taken modulo n.
- A103134 (program): a(n) = Fibonacci(6n+4).
- A103136 (program): Inverse of the Delannoy triangle.
- A103137 (program): First column of inverse of Delannoy triangle.
- A103138 (program): Second column of inverse of Delannoy triangle.
- A103142 (program): a(n) = 2*a(n-1) + a(n-2) + a(n-3) + a(n-4), with a(0)=1, a(1)=2, a(3)=5, a(4)=13.
- A103143 (program): a(n) = a(n-1) + a(n-2) + 3*a(n-3), with a(0) = 1, a(1) = 0, a(2) = 1.
- A103144 (program): Decimal primes whose decimal representation in hex is also prime.
- A103145 (program): a(n) = (1/6)*(n^3 + 21*n^2 + 74*n + 18).
- A103151 (program): Number of decompositions of 2n+1 into 2p+q, where p and q are both odd primes (A065091).
- A103154 (program): Each letter appears an even number of times in the English names for 1 through n taken together (names without “and”).
- A103157 (program): Number of ways to choose 4 distinct points from an (n+1) X (n+1) X (n+1) lattice cube.
- A103159 (program): a(n) = GCD(reverse(n), reversed(n+1)).
- A103160 (program): a(n) = GCD(reverse(n!), reverse((n+1)!)).
- A103161 (program): GCD of reverse(2^n) and reverse(2^(n+1)), where reverse(k) = A004086(k), the decimal representation of k read backwards.
- A103162 (program): GCD of reverse(3^n) and reverse(3^(n+1)).
- A103164 (program): Integers but with the primes squared.
- A103166 (program): a(n) = reverse(2^n) mod 2^n.
- A103167 (program): a(n) = 2^n mod reverse(2^n).
- A103168 (program): a(n) is the remainder when (n written backwards) is divided by n.
- A103175 (program): A001787 written in base 2.
- A103177 (program): (7*3^n + 2n + 5)/4.
- A103181 (program): In decimal representation of n: replace all even digits with 0 and all odd digits with 1.
- A103185 (program): a(n) = Sum_{ k >= 0 such that n + k == 0 mod 2^k } 2^(k-1).
- A103187 (program): a(n) = second term in continued fraction of n-th harmonic number.
- A103188 (program): Numbers k such that k711 is prime.
- A103190 (program): Numbers k such that k29 is prime.
- A103192 (program): Trajectory of 1 under repeated application of the function n -> A102370(n).
- A103194 (program): LAH transform of squares.
- A103196 (program): a(n) = (1/9)(2^(n+3)-(-1)^n(3n-1)).
- A103197 (program): Number of compositions of n into Fibonacci number of parts.
- A103198 (program): Number of compositions of n into a square number of parts.
- A103200 (program): a(1)=1, a(2)=2, a(3)=11, a(4)=19; a(n) = a(n-4) + sqrt(60*a(n-2)^2 + 60*a(n-2) + 1) for n >= 5.
- A103201 (program): a(1) = 11, a(2) = 19, a(3) = 89, a(4) = 151; for n >= 5, a(n) = sqrt(a(n-4)^2 + 60*a(n-2)^2 + 4*a(n-2)*sqrt(210 + 15*a(n-4)^2)).
- A103202 (program): A102370 sorted.
- A103204 (program): a(1) = 2, a(2) = 4; a(n) = 2*a(n-1) - 1.
- A103205 (program): Write numbers in decimal under each other, then read diagonals in upward direction.
- A103208 (program): Numbers n such that 3 divides prime(1) + … + prime(n).
- A103210 (program): a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*2^i*3^(n-i), a(0)=1.
- A103211 (program): a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*3^i*4^(n-i), a(0)=1.
- A103212 (program): a(n) = (1/n) * Sum_{i=0..n-1} C(n,i)*C(n,i+1)*(n-1)^i*n^(n-i) for n>=1, a(0)=1.
- A103213 (program): a(n) = n! * Sum_{k=1..n} binomial(n,k)/k.
- A103214 (program): a(n) = 24*n + 1.
- A103215 (program): Numbers congruent to {1, 2, 5, 10, 13, 17} mod 24.
- A103217 (program): Hexagonal numbers triangle read by rows: T(n,k)=(n+1-k)*(2*(n+1-k)-1).
- A103218 (program): Triangle read by rows: T(n, k) = (2*k+1)*(n+1-k)^2.
- A103219 (program): Triangle read by rows: T(n,k) = (n+1-k)*(4*(n+1-k)^2 - 1)/3+2*k*(n+1-k)^2.
- A103220 (program): a(n) = n*(n+1)*(3*n^2+n-1)/6.
- A103221 (program): Number of partitions of n into parts 2 and 3.
- A103247 (program): Triangle read by rows: T(n,k) is the coefficient of x^k (0<=k<=n) in the monic characteristic polynomial of the n X n matrix with 3’s on the diagonal and 1’s elsewhere (n>=1). Row 0 consists of the single term 1.
- A103252 (program): Array A000292(n)*A000217(k), read by antidiagonals.
- A103260 (program): Number of partitions of 2n prime to 3 with all odd parts occurring with multiplicity 2. The even parts occur with multiplicity 1.
- A103262 (program): McKay-Thompson series of class 36g for the Monster group.
- A103271 (program): a(n) = (prime(n)+prime(n+1)) mod 4.
- A103273 (program): Number of ways of writing prime(n)-1 in the form prime(i)+prime(j).
- A103279 (program): Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1];.
- A103280 (program): Array read by antidiagonals, generated by the matrix M = [1,1,1;1,N,1;1,1,1];.
- A103283 (program): Triangle read by rows: T(n,k) is the coefficient of x^k in the monic characteristic polynomial of the n X n matrix with 2’s on the diagonal and 1’s elsewhere (n >= 1 and 0 <= k <= n). Row 0 consists of the single term 1.
- A103288 (program): Numbers k such that sigma(k) >= 2k-1 (union of perfect, abundant and least deficient numbers).
- A103290 (program): n*(n-1)*(n^2-n+4)/6.
- A103303 (program): Complete list of digits used in the counting numbers (in base 10). Also known as the “Arabic numerals”.
- A103311 (program): A transform of the Fibonacci numbers.
- A103312 (program): A transform of the Jacobsthal numbers.
- A103316 (program): Riordan array (1/(1+2x), x/(1+x)).
- A103318 (program): Number of solutions i in range [0,n-1] to i == 0 mod 2^(n-i).
- A103321 (program): Expansion of 1 / ((1-x-x^2-x^3)*(1-x-x^3)).
- A103322 (program): Expansion of 1 / ((1-x-x^2-x^3)*(1-x^2-x^3)).
- A103323 (program): Square array T(n,k) read by antidiagonals: powers of Fibonacci numbers.
- A103324 (program): Square array T(n,k) read by antidiagonals: powers of Lucas numbers.
- A103325 (program): Fifth powers of Lucas numbers.
- A103326 (program): a(n) = Fibonacci(5n)/Fibonacci(n).
- A103327 (program): Triangle read by rows: T(n,k) = binomial(2n+1, 2k+1).
- A103328 (program): Triangle T(n, k) read by rows: binomial(2n, 2k+1).
- A103333 (program): Number of closed walks on the graph of the (7,4) Hamming code.
- A103334 (program): Number of closed walks of length 2n at any of the nodes of degree 1 on the graph of the (7,4) Hamming code.
- A103340 (program): Denominator of the unitary harmonic mean (i.e., the harmonic mean of the unitary divisors) of the positive integer n.
- A103345 (program): Numerator of Sum_{k=1..n} 1/k^6 = Zeta(6,n).
- A103346 (program): Denominators of Sum_{k=1..n} 1/k^6 = Zeta(6,n).
- A103347 (program): Numerators of Sum_{k=1..n} 1/k^7 = Zeta(7,n).
- A103348 (program): Denominators of sum_{k=1..n} 1/k^7 = Zeta(7,n).
- A103349 (program): Numerators of sum_{k=1..n} 1/k^8 = Zeta(8,n).
- A103350 (program): Denominators of sum_{k=1..n} 1/k^8 = Zeta(8,n).
- A103351 (program): Numerators of sum_{k=1..n} 1/k^9 = Zeta(9,n).
- A103352 (program): Denominators of sum_{k=1..n} 1/k^9 = Zeta(9,n).
- A103354 (program): a(n) = floor(x), where x is the solution to x = 2^(n-x).
- A103355 (program): a(n) = n - floor( sqrt(prime(n) ).
- A103359 (program): T(n,k) = Max{p: prime p divides m where n-k<=m<=n+k}, triangle read by rows: 0<=k<n.
- A103368 (program): Period 6: repeat [1, 1, -1, -1, 0, 0].
- A103370 (program): Row sums of triangle A095801 (matrix square of the Narayana triangle A001263).
- A103371 (program): Number triangle T(n,k) = C(n,n-k)*C(n+1,n-k).
- A103372 (program): a(1) = a(2) = a(3) = a(4) = a(5) = 1 and for n>5: a(n) = a(n-4) + a(n-5).
- A103373 (program): a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1 and for n>6: a(n) = a(n-5) + a(n-6).
- A103374 (program): a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = 1 and for n>7: a(n) = a(n-6) + a(n-7).
- A103375 (program): a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = 1 and for n>8: a(n) = a(n-7) + a(n-8).
- A103376 (program): a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = 1 and for n>9: a(n) = a(n-8) + a(n-9).
- A103377 (program): a(1)=a(2)=…=a(10)=1, a(n)=a(n-9)+a(n-10).
- A103379 (program): k=11 case of family of sequences beyond Fibonacci and Padovan.
- A103380 (program): k=12 case of family of sequences beyond Fibonacci and Padovan: a(n) = a(n-12) + a(n-13).
- A103390 (program): Natural numbers but with nonprimes squared.
- A103391 (program): ‘Even’ fractal sequence for the natural numbers: Deleting every even-index term results in the same sequence.
- A103401 (program): Numbers k such that k211 is prime.
- A103406 (program): Triangle read by rows: n-th row = unsigned coefficients of the characteristic polynomials of an n X n matrix with 2’s on the diagonal and 1’s elsewhere.
- A103407 (program): Triangle of absolute values of the coefficients (in descending powers) of the characteristic polynomials of n X n matrices with 3’s on the main diagonal and 1’s elsewhere.
- A103408 (program): Numbers n such that n2101 is prime.
- A103409 (program): Numbers n such that n2357 is prime.
- A103410 (program): Number of products of distinct elements in generation n, starting with two elements.
- A103416 (program): a(n) = n - ceiling(sqrt(prime(n))).
- A103419 (program): Number of compositions of n in which the least part is odd.
- A103420 (program): Number of compositions of n in which the least part is even.
- A103424 (program): Expansion of e.g.f.: 1 + sinh(2*x).
- A103425 (program): a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3).
- A103433 (program): a(n) = Sum_{i=1..n} Fibonacci(2i-1)^2.
- A103434 (program): a(n) = Sum_{i=1..n} Fibonacci(2i)^2.
- A103435 (program): a(n) = 2^n * Fibonacci(n).
- A103438 (program): Square array T(m,n) read by antidiagonals: Sum_{k=1..n} k^m.
- A103439 (program): a(n) = Sum_{i=0..n-1} Sum_{j=0..i} (i-j+1)^j.
- A103440 (program): a(n) = Sum[d|n, d==1 (mod 3), d^2] - Sum[d|n, d==2 (mod 3), d^2].
- A103444 (program): Triangle read by rows: T(n,k) is number of unitary divisors of C(n,k), 0<=k<=n.
- A103445 (program): Sum of the numbers of unitary divisors of the binomial coefficients C[n,k], k=0..n.
- A103446 (program): Unlabeled analog of A025168.
- A103447 (program): Triangle read by rows: T(n,k) = Moebius(binomial(n,k)) (0 <= k <= n).
- A103448 (program): a(n) = Sum_{k=0..n} Moebius(binomial(n,k)).
- A103450 (program): A figurate number triangle read by rows.
- A103451 (program): Triangular array T read by rows: T(n, 0) = T(n, n) = 1, T(n, k) = 0 for 0 <= k <= n.
- A103452 (program): Inverse of number triangle A103451.
- A103453 (program): a(n) = 0^n + 3^n - 1.
- A103454 (program): a(n) = 0^n + 4^n - 1.
- A103455 (program): a(n) = 0^n + 5^n - 1.
- A103456 (program): a(n) = 0^n + 10^n - 1.
- A103457 (program): a(n) = 3^n + 1 - 0^n.
- A103458 (program): a(n) = 7^n + 1 - 0^n.
- A103459 (program): a(n) = 8^n + 1 - 0^n.
- A103460 (program): a(n) = 9^n + 1 - 0^n.
- A103461 (program): a(n) = (-10)^n + 1 - 0^n.
- A103462 (program): A triangle with palindromic cubes, read by rows.
- A103469 (program): Number of polyominoes consisting of 3 regular unit n-gons.
- A103480 (program): Row sums of A103462.
- A103481 (program): Diagonal sums of A103462.
- A103485 (program): Expansion of (7 +4*x -5*x^2 -7*x^3) / ((1-x)*(1-x^2-x^3)).
- A103486 (program): a(0)=7, a(1)=11, a(2)=13, a(3)=17; then a(n) = a(n-1)+a(n-3)-a(n-4).
- A103488 (program): a(n) = 2^(n^2-1).
- A103489 (program): Multiplicative suborder of 3 (mod n) = sord(3, n).
- A103491 (program): Multiplicative suborder of 5 (mod n) = sord(5, n).
- A103492 (program): Multiplicative suborder of 6 (mod 2n+1) = sord(6, 2n+1).
- A103493 (program): Multiplicative suborder of 7 (mod n) = sord(7, n).
- A103495 (program): Multiplicative suborder of 9 (mod n) = sord(9, n).
- A103496 (program): Multiplicative suborder of 10 (mod 2n+1) = sord(10, 2n+1).
- A103497 (program): Multiplicative suborder of 11 (mod n) = sord(11, n).
- A103498 (program): Multiplicative suborder of 12 (mod 2n+1) = sord(12, 2n+1).
- A103505 (program): Denominator in expansion of (1-x)*log(1-x).
- A103516 (program): Triangle read by rows: count in a vee.
- A103517 (program): Expansion of (1+2*x-x^2)/(1-x)^2.
- A103519 (program): a(1) = 1, a(n) = sum of n successive numbers starting with a(n-1) + 1.
- A103528 (program): Sum_{k = 1..n-1 such that n == k (mod 2^k)} 2^(k-1).
- A103529 (program): Values of A102370 which are >= a new power of 2.
- A103530 (program): a(n) = 2^n - A103529(n).
- A103532 (program): Number of divisors of 240^n.
- A103534 (program): Concatenations of pairs of primes that differ by 1000.
- A103536 (program): Number of nets in a regular pyramid.
- A103542 (program): Binary equivalents of A102370.
- A103543 (program): Consider those values of k for which A102370(k) = k: 0, 4, 8, 16, 20, 24, 32, 36, 40, 48, 52, 56, 64, … and divide by 4: 0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 20, 21, 22, 24, …; sequence gives missing numbers.
- A103566 (program): Sum of the primes > 5 modulo 3.
- A103567 (program): Sum of the (primes > 5 modulo 5).
- A103568 (program): Sum of the (primes > 5 modulo 7).
- A103569 (program): Sum of the (primes > 5 modulo 11).
- A103570 (program): Sum of the (primes > 5 modulo 13).
- A103571 (program): Sum of the (primes > 5 modulo 17).
- A103572 (program): Sum of the (primes > 5 modulo 19).
- A103577 (program): Number of partitions of n into Fibonacci parts if each part is of two kinds.
- A103578 (program): Number of divisors of m^2, where m is the smallest number with at least n divisors.
- A103579 (program): Sophie Germain primes that are not Lucasian primes: primes p not 3 (mod 4) such that 2p + 1 is prime.
- A103581 (program): A102371 written in base 2.
- A103582 (program): Binary array below read by downward antidiagonals.
- A103583 (program): Same as A103582, but read antidiagonals in upward direction.
- A103585 (program): Consider numbers k such that (A102370(k)-k)/2 = 1; read them mod 4 to get the sequence.
- A103586 (program): a(0)=1, for n > 0: n-th run consists of 2^n-1 copies of n+1.
- A103588 (program): 1’s complement of A103582.
- A103589 (program): 1’s complement of A103583.
- A103601 (program): Numbers k such that the string 10k is the decimal expansion of a prime number.
- A103604 (program): a(n) = C(n+6,6) * C(n+10,6).
- A103609 (program): Fibonacci numbers repeated (cf. A000045).
- A103610 (program): 4 X infinity array read by rows: let M = {{0, 1, 0, 0}, {0, 0, 1, 0}, {0, 0, 0, 1}, {1, 0, 1, 0}}, w[0] = {0, 0, 1, 1}’, w[n]’ = M*w[n -1]’; the n-th row of the array is w[n]’ (the prime denotes transpose).
- A103612 (program): Number of solutions to 5+B^2=p^2+q^2 with B=2n, p,q>0 and 2p^2<5+B^2.
- A103615 (program): Number of zeros in A103542(n) (binary equivalent of A102370(n)).
- A103616 (program): Decimal expansion of the largest real root of the quintic equation x^5 + 2*x^4 - 2*x^3 - x^2 + 2*x -1 = 0.
- A103621 (program): Trajectory of 7 under repeated application of the map n –> A102370(n).
- A103623 (program): n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1.
- A103625 (program): Define a(1)=0, a(2)=0, a(3)=2, a(4)=4, a(5)=34, a(6)=62, a(7)=480, a(8)=870 such that from i=1 to 8: 48*a(i)^2 + 48*a(i) + 1 = j(i)^2 with j(1)=1, j(2)=1, j(3)=17, j(4)=31, j(5)=239, j(6)=433, j(7)=3329, j(8)=6031. Then a(n) = a(n-8) + 28*sqrt(48*(a(n-4)^2) + 48*a(n-4) + 1).
- A103627 (program): Let S(n) = {n,1,n}; sequence gives concatenation S(0), S(1), S(2), …
- A103631 (program): Triangle read by rows: T(n,k) = abs(qStirling2(n,k,q)) for q = -1, with 0 <= k <= n.
- A103632 (program): Expansion of (1 - x + x^2)/(1 - x - x^4).
- A103633 (program): Triangle read by rows: triangle of repeated stepped binomial coefficients.
- A103636 (program): Sum[d|n, d==0 mod 3, d^2].
- A103637 (program): Sum[d|n, d==1 mod 3, d^2].
- A103638 (program): Sum[d|n, d==2 mod 3, d^2].
- A103639 (program): a(n) = Product_{i=1..2*n} (2*i+1).
- A103640 (program): Expansion of theta_4(q)^4 - theta_2(q)^4, where theta_2 and theta_4 are the Jacobi theta series.
- A103644 (program): Expansion of g.f. (3x+1)/(1+2x-6x^2-27x^3).
- A103645 (program): G.f.: (108x^2+27x+1)/(1+2x-6x^2-27x^3).
- A103646 (program): G.f.: 9*(3x+1)/(1+2x-6x^2-27x^3).
- A103650 (program): G.f.: x^2/((1-x^2)^2*Product_{i>0}(1-x^i)).
- A103659 (program): (1/6) * most frequently occurring volume assumed by triangular pyramids with their 4 vertices chosen from distinct points of an (n+1)X(n+1)X(n+1) lattice cube.
- A103668 (program): Number of semiprimes between prime(n) and prime(n+1).
- A103675 (program): a(n) = 1 if the binary representation of n! contains 7! (bit string “1001110110000”), otherwise a(n) = 0.
- A103681 (program): Numbers m such that in binary representation m! doesn’t contain 7!.
- A103685 (program): Consider the morphism 1->{1,2}, 2->{1,3}, 3->{1}; a(n) is the total number of ‘3’ after n substitutions.
- A103689 (program): a(n) is the least k such that either k*n - 1 or k*n + 1 (or both) is prime.
- A103701 (program): Add 2 to each of the preceding digits, beginning with 1.
- A103704 (program): Add 5 to each of the preceding digits, beginning with 1.
- A103716 (program): Numerators of sum_{k=1..n} 1/k^10 =: Zeta(10,n).
- A103717 (program): Denominators of sum_{k=1..n} 1/k^10 = Zeta(10,n).
- A103719 (program): Column m=2 sequence of triangle A103718(n,m), n >= 0 (without leading zeros).
- A103720 (program): Column m=3 sequence (unsigned) of triangle A103718(n,m), n>=0, without leading zeros.
- A103721 (program): Column m=4 sequence of triangle A103718(n,m), n>=0, without leading zeros.
- A103722 (program): Column m=5 sequence of triangle A103718(n,m), n>=0, without leading zeros.
- A103729 (program): Column k=2 sequence of array A103728.
- A103730 (program): Negative of column k=3 sequence of array A103728.
- A103731 (program): Column k=4 sequence of array A103728.
- A103736 (program): Fibonacci numbers with nonprime indices.
- A103737 (program): Define a(1)=0, a(2)=0, a(3)=3, a(4)=7 such that from i=1 to 4: 30*a(i)^2 + 30*a(i) + 1 = j(i)^2, j(1)=1, j(2)=1, j(3)=19, j(4)=41 Then a(n) = a(n-4) + 4*sqrt(30*(a(n-2)^2) + 30*a(n-2) + 1).
- A103745 (program): a(n) = (A102371(n) + n)/2.
- A103747 (program): Trajectory of 2 under repeated application of the map n -> A102370(n).
- A103749 (program): Expansion of x*(1+2*x)/(1+x+x^2-2*x^3).
- A103750 (program): Expansion of (1+2*x^3)/(1-x+x^3-2*x^4).
- A103754 (program): Number of contiguous digits i in the counting numbers, for i=0.
- A103762 (program): a(n) = least k with Sum_{j = n..k} 1/j >= 1.
- A103768 (program): (29*3^n - 18(n + 3)*2^n + 6n^2 + 24n + 27)/12.
- A103770 (program): A weighted tribonacci sequence, (1,3,9).
- A103771 (program): Expansion of 1/(1-4x-4x^2-4x^3).
- A103772 (program): Larger of two sides in a (k,k,k-1)-integer-sided triangle with integer area.
- A103775 (program): Number of ways to write n! as product of distinct squarefree numbers.
- A103778 (program): Inverse of trinomial triangle A071675.
- A103779 (program): Expansion of real root of y + y^2 + y^3 = x.
- A103781 (program): Sum of any four successive terms is prime, a(1)=a(2)=0,a(3)=1.
- A103796 (program): Indices of n such that A019565(n)+1 is prime.
- A103797 (program): Indices of n such that A019565(n)-1 is prime.
- A103799 (program): Indices n such that A019565(n)+2 is prime.
- A103813 (program): Partial sums of A102370.
- A103814 (program): Pentanacci constant: decimal expansion of limit of A001591(n+1)/A001591(n).
- A103815 (program): a(n) = -1 + Product_{k=1..n} Fibonacci(k).
- A103816 (program): Numerator of Sum_{k=1..n} (-1)^(k+1)/k!.
- A103819 (program): Whitney transform of Jacobsthal numbers.
- A103820 (program): Whitney transform of 3^n.
- A103821 (program): A Whitney transform of the central binomial coefficients A000984.
- A103826 (program): Unitary arithmetic numbers (those for which the arithmetic mean of the unitary divisors is an integer).
- A103831 (program): For even n, a(n) = n*(n+1), for odd n, a(n) = 2*n + 1.
- A103832 (program): For even n, a(n)=2n+1, for odd n, a(n)=n(n+1)
- A103838 (program): Complement of A001671.
- A103842 (program): Triangle read by rows: row n is binary expansion of 2^n-n, n >= 1.
- A103845 (program): Product of first n Lucas numbers, plus one.
- A103847 (program): McCarthy’s 91 Function: a(n) = n-10 if n>100, otherwise a(n) = a(a(n+11)).
- A103848 (program): Numbers n such that sum of even digits of n is larger than sum of odd digits.
- A103855 (program): a(n) = Prime(n)! - prime(n)# + 1.
- A103863 (program): Hamming distance between n and A102370(n) (in binary).
- A103868 (program): Digital expansion of Pi: numbers from each pair of successive digits, reversed.
- A103871 (program): Numbers n such that 100n + 69 is prime.
- A103872 (program): a(n) = 3*trinomial(n+1,0) - trinomial(n+2,0).
- A103876 (program): A test for divisibility by the n-th prime p(n).
- A103881 (program): Square array T(n,k) (n >= 1, k >= 0) read by antidiagonals: coordination sequence for root lattice A_n.
- A103882 (program): a(n) = Sum_{i=0..n} C(n+1,i)*C(n-1,i-1)*C(2n-i,n).
- A103884 (program): Square array T(n,k) read by antidiagonals: coordination sequence for lattice C_n.
- A103885 (program): a(n) = [x^(2*n)] ((1 + x)/(1 - x))^n.
- A103889 (program): Odd and even positive integers swapped.
- A103890 (program): Prime(n)! / prime(n)# + 1.
- A103897 (program): a(n) = 3*2^(n-1)*(2^n-1).
- A103904 (program): a(n) = n*(n-1)/2 * 2^(n*(n-1)/2).
- A103916 (program): Column k=2 sequence (without zero entries) of table A060524.
- A103930 (program): Numerators of squares of harmonic numbers A001008/A002805.
- A103931 (program): Denominators of squares of harmonic numbers A001008/A002805.
- A103932 (program): Numerators of first difference of squares of harmonic numbers.
- A103933 (program): Denominators of first difference of squares of harmonic numbers A001008/A002805.
- A103938 (program): Number of rooted non-separable n-edge maps in the plane (planar with a distinguished outside face).
- A103943 (program): Number of unrooted two-vertex n-edge maps in the plane (planar with a distinguished outside face).
- A103944 (program): Number of rooted unicursal n-edge maps in the plane (planar with a distinguished outside face).
- A103947 (program): a(n) is the number of distinct n-th powers of functions {1, 2} -> {1, 2}.
- A103951 (program): Procedure “Remove every 10th term!” executed 10 times.
- A103961 (program): Least k such that 2*n*k - 1 is a prime.
- A103969 (program): Positions n such that A005941(n) = A005940(n).
- A103970 (program): Expansion of (1 - sqrt(1 - 4*x - 12*x^2))/(2*x).
- A103971 (program): Expansion of (1 - sqrt(1 - 4x - 16x^2))/(2x).
- A103972 (program): Expansion of (1-sqrt(1-4x-20x^2))/(2x).
- A103973 (program): Expansion of (sqrt(1-8*x^2)+8*x^2+2*x-1)/(2*x*sqrt(1-8*x^2)).
- A103974 (program): Smaller sides (a) in (a,a,a+1)-integer triangle with integer area.
- A103975 (program): Smaller side in (a,a+1,a+1)-integer triangle with integer area.
- A103976 (program): Partial sums of A040976 (= primes-2).
- A103977 (program): Let d_1 … d_k be the divisors of n. Then a(n) = min_{ e_1 = +-1, … e_k = +-1 } | Sum_i e_i d_i |.
- A103978 (program): Expansion of (sqrt(1-12*x^2)+12*x^2+2*x-1)/(2*x*sqrt(1-12*x^2)).
- A103981 (program): Number of prime factors (with multiplicity) of octahedral numbers (A005900).
- A103982 (program): Indices of octahedral numbers (A005900) which are semiprimes.
- A103990 (program): Reduced numerators of the hypercube line-picking integrand sqrt(Pi)*I(n,0).
- A103991 (program): Reduced denominators of the hypercube line-picking integrand sqrt(Pi)*I(n,0).
- A103994 (program): A129360 * A115361.
- A103996 (program): Recurrence: a(n) = -Sum[i=0..n-1, a(i)*C(n+1,i) ], a(0)=1.
- A104000 (program): Square array T(r,m) read by antidiagonals: number of cyclically reduced words of length m in F_r.
- A104001 (program): Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns starting with fixed m, 2<k<=n, 1<=m<=k.
- A104002 (program): Triangle T(n,k) read by rows: number of permutations in S_n avoiding all k-length patterns that start with 1 except one fixed pattern and containing it exactly once.
- A104004 (program): Expansion of (1-x)(1+x)/((2x-1)(x^2+x-1)).
- A104005 (program): a(n+3) = a(n+2) + 3a(n+1) - 2a(n); a(0) = 1, a(1) = -1, a(2)= 3.
- A104006 (program): Primes of the form 2pq + 1, where p and q are (not necessarily distinct) odd primes.
- A104008 (program): Area of (m,m+1,m+1)-integer triangle for m in A103975.
- A104009 (program): Area of (a,a,a+1)-integer triangle. Corresponding a’s are in A103974.
- A104010 (program): Sum of two successive sexy primes.
- A104011 (program): Number of prime factors (with multiplicity) of centered dodecahedral numbers (A005904).
- A104012 (program): Indices of centered dodecahedral numbers (A005904) which are semiprimes (A001358).
- A104029 (program): Triangle, read by rows, of pairwise sums of trinomial coefficients (A027907).
- A104033 (program): Triangle, read by rows, equal to the matrix inverse of triangle A103327, where A103327(n,k) = binomial(2*n+1,2*k+1).
- A104039 (program): Number of primitive roots modulo prime(n)^2, where prime(n) is n-th prime.
- A104040 (program): Triangular matrix T, read by rows, such that row n equals the absolute values of column (n+1) in the matrix inverse T^-1 (with extrapolated zeros): T(n,k) = -Sum_{j=1..[n+1/2]} (-1)^j*T(n-j+1,n-2*j+1)*T(n-j,k)) with T(n,n)=1 (n>=0) and T(n,n-1)=n (n>=1).
- A104041 (program): Triangular matrix T, read by rows, such that column k is equal (in absolute value) to row (k-1) in the matrix inverse T^-1 (with extrapolated zeros) for k>0, with T(n,n)=1 (n>=0) and T(n,n-1)=-n (n>=1).
- A104044 (program): Numbers k such that 10*k + 7 is prime and k is a multiple of ten.
- A104045 (program): Numbers n such that n9 is prime and n is a multiple of ten.
- A104048 (program): Numbers n such that n11 is prime and n is a multiple of ten.
- A104055 (program): Number of numbers 0 <= i <= n such that i is a square or a cube (or both).
- A104078 (program): Numbers which are the sum of three nonzero squares and are also divisible by 31.
- A104085 (program): Coefficient list length of Poincaré-like polynomials made from A047845, indices of odd nonprimes (group dimension equivalent plus one).
- A104088 (program): Largest prime <= 3^n.
- A104097 (program): Denominators of coefficients in expansion of x^-2*(1-exp(-2*x))^2.
- A104098 (program): Sum_{k=1..n} C(n-1,k-1)*A008292(n,k) for n>=1.
- A104099 (program): n * (10n^2 - 6n + 1), or n*A087348(n).
- A104103 (program): a(n) = ceiling(sqrt(prime(n))).
- A104104 (program): a(1) = 1, if A(k) = sequence of first 2^(k-1) terms and if B(k) is A(k) with 0’s and 1’s exchanged, then A(k+1) = A(k)A(k) if a(k) = 0, A(k+1) = A(k)B(k) if a(k) = 1.
- A104105 (program): a(1) = 1, if A(k) = sequence of first 2^k -1 terms and if B(k) is A(k) with 0’s and 1’s exchanged, then A(k+1) = A(k),1,B(k) if a(k) = 0, A(k+1) = A(k),0,B(k) if a(k) = 1.
- A104106 (program): a(1) = 1; thereafter, if A(k) = sequence of first 2^k -1 terms, then A(k+1) = A(k),1,A(k) if a(k) = 0, and A(k+1) = A(k),0,A(k) if a(k) = 1.
- A104110 (program): Nonnegative numbers k such that k^2 + 42 is prime.
- A104117 (program): For n=2^k, a(n) = k+1, else 0.
- A104120 (program): (Prime(n + 1) - Prime(n))/2 (mod 2).
- A104121 (program): a(n)=1 if there is a partition of n^3 into n primes such that n-1 are consecutive primes and the remaining prime is larger than the sum of the n-1 consecutive primes, otherwise a(0)=0 if no such partition exists.
- A104127 (program): (1+prime(n))^prime(n).
- A104128 (program): a(n) = p + p^(p+1), where p = prime(n).
- A104129 (program): Integers of the form p^(p-1)+p where p is prime.
- A104130 (program): Numbers n such that n33 is prime and n is a multiple of ten.
- A104135 (program): a(n) = floor(sqrt(2*Pi*n)).
- A104136 (program): a(n) = ceiling(sqrt(2*Pi*n)).
- A104137 (program): Number of distinct necklaces with p beads of two possible colors, allowing turning over, p being a prime greater than 2.
- A104141 (program): Decimal expansion of 3/Pi^2.
- A104144 (program): a(n) = Sum_{k=1..9} a(n-k); a(8) = 1, a(n) = 0 for n < 8.
- A104145 (program): a(1) = 1; let A(k) = sequence of first 2^(k-1) terms; then A(k+1) is concatenation of A(k) and (A(k)-1) if a(k) is odd, or concatenation of A(k) and (A(k)+1) if a(k) is even.
- A104147 (program): Number of cubes <= n-th prime.
- A104150 (program): Shifted factorial numbers: a(0)=0, a(n) = (n-1)!.
- A104152 (program): Numbers n such that n77 is prime and n is a multiple of ten.
- A104153 (program): Numbers n such that n99 is prime and n is a multiple of ten.
- A104155 (program): The 64 codons of the genetic code, giving the value 1 to thymine (T), 3 to adenine (A), 2 to cytosine (C) and 4 to guanine (G).
- A104156 (program): a(1)=a(2)=0, a(n) = abs(2*a(n-1) - a(n-2)) - 1.
- A104161 (program): G.f.: x*(1 - x + x^2)/((1-x)^2 * (1 - x - x^2)).
- A104162 (program): Indicator sequence for the Fibonacci numbers.
- A104163 (program): Primes p such that (2p+1)/3 is prime.
- A104164 (program): Sophie Germain type primes where 5*Prime[n]=2*Prime[m]+1.
- A104165 (program): Sophie Germain type primes where 7*Prime[n]=2*Prime[m]+1.
- A104174 (program): Numerator of the fractional part of a harmonic number.
- A104175 (program): From the words to the song “867-5309/Jenny” by Tommy Tutone.
- A104177 (program): A variation on Flavius’s sieve (A000960): Start with the natural numbers; at the k-th sieving step, remove every f-th term of the sequence remaining after the (k-1)-st sieving step, where f is the (k+2)-nd Fibonacci number, f=F(k+2); iterate.
- A104180 (program): Let f[n]=Prime[n+1]-Prime[n]; a(n) = Binomial[Prime[12],f[n]].
- A104181 (program): Let f(n)=mod(prime(n),12); then a(n) = binomial(prime(12),f(n)).
- A104184 (program): a(n) is the number of paths from (0,0) to (n,0) using steps of the form (1,2),(1,1),(1,0),(1,-1) or (1,-2) and staying above the x-axis. Also, a(n) is the number of possible combinations of balls on the lawn after n turns, using a Motzkin variation of the (4,2)-case of the tennis ball problem considered by D. Merlini, R. Sprugnoli and M. C. Verri.
- A104187 (program): G.f. -(1+x^2+x^4)/((x^3+x^2+x-1)*(x-1)^2).
- A104188 (program): a(n) = 4n*(4n - 1).
- A104189 (program): Prime numbers arising from Schorn’s proof that there are infinitely many primes.
- A104192 (program): a(n) = prime(n) - phi(n).
- A104199 (program): Lower bound on a straddle prime pair.
- A104200 (program): Upper bound on a straddle prime pair.
- A104201 (program): Sums of straddle primes.
- A104202 (program): Differences of straddle primes.
- A104210 (program): Positive integers divisible by at least 2 consecutive primes.
- A104211 (program): Integers n such that the sum of the digits of n is not prime.
- A104212 (program): Sum of the digits of n when the sum is prime.
- A104213 (program): Primes with nonprime sums of digits.
- A104218 (program): Sum of opposite numbers on a clock, starting at 12.
- A104219 (program): Triangle read by rows: T(n,k) is number of Schroeder paths of length 2n and having k peaks at height 1, for 0 <= k <= n.
- A104220 (program): a(n) = 1 + Fibonacci(n) - (Fibonacci(n) mod 2).
- A104221 (program): a(n) = Fibonacci(n) - (Fibonacci(n) mod 2).
- A104230 (program): Minimal number of primes needed to sum to n^2.
- A104234 (program): Number of k >= 1 such that k+n == 0 mod 2^k.
- A104235 (program): Numbers n such that A102370(n) = n.
- A104236 (program): n*Golomb’s sequence.
- A104248 (program): Lengths of successive runs of 1’s in the Thue-Morse sequence A010060.
- A104249 (program): a(n) = (3*n^2 + n + 2)/2.
- A104250 (program): Sum of prime digits of n-th prime.
- A104251 (program): Sum of nonprime digits of n-th prime.
- A104254 (program): n^n - (-1)^n(n+1)!.
- A104255 (program): a(n) = floor( (2n-1)!!/(2n) ).
- A104256 (program): [ (2n-1)!!/(4n) ].
- A104258 (program): Replace 2^i with n^i in binary representation of n.
- A104259 (program): Triangle T read by rows: matrix product of Pascal and Catalan triangle.
- A104260 (program): Sum of odd digits (1,3,5,7,9) of n-th prime.
- A104261 (program): Sum of even digits (0,2,4,6,8) of n-th prime.
- A104268 (program): a(n) = 2*4^(n-1) - (3n-1)/(2n+2)*C(2n,n).
- A104270 (program): a(n) = 2^(n-2)*(C(n,2)+2).
- A104273 (program): Table of Sprague-Grundy functions for a certain family of hypergraphs, read by antidiagonals.
- A104275 (program): Numbers k such that 2k-1 is not prime.
- A104276 (program): Number of partitions of n in which both even and odd squares occur with multiplicity 1. There is no restriction on the parts which are twice even squares.
- A104278 (program): Numbers n such that 2n+1 and 2n-1 are not primes.
- A104279 (program): Numbers n such that 2n+1 is prime and 2n-1 is not prime.
- A104280 (program): Numbers n such that 2n+1 is not prime and 2n-1 is prime.
- A104293 (program): a(n) = prime((prime(n)-1)/2).
- A104294 (program): a(n) = prime((prime(n)+1)/2).
- A104295 (program): a(n) = A104294(n) - A104293(n).
- A104301 (program): Primes which are the reverse concatenation of two consecutive square numbers.
- A104320 (program): Number of zeros in ternary representation of 2^n.
- A104324 (program): The Fibonacci word over the nonnegative integers; or, the number of runs of identical bits in the binary Zeckendorf representation of n.
- A104325 (program): Number of runs of equal bits in the dual Zeckendorf representation of n (A104326).
- A104326 (program): Dual Zeckendorf representation of n or the maximal (binary) Fibonacci representation. Also list of binary vectors not containing 00.
- A104332 (program): Primes which are the reverse concatenation of two consecutive odd numbers.
- A104344 (program): a(n) = Sum_{k=1..n} k!^2.
- A104350 (program): Partial products of largest prime factors of numbers <= n.
- A104351 (program): Number of digits in decimal representation of A104350(n).
- A104352 (program): Number of divisors of A104350(n).
- A104354 (program): Euler’s totient of A104350(n).
- A104355 (program): Number of trailing zeros in decimal representation of A104350(n).
- A104356 (program): Smallest m such that A104350(m) has exactly n trailing zeros in decimal representation.
- A104357 (program): a(n) = A104350(n) - 1.
- A104365 (program): a(n) = A104350(n) + 1.
- A104376 (program): a(n) = Sum_{j=0..13} n^j.
- A104378 (program): First differences of A102370.
- A104381 (program): Numbers k such that 10^(k-1) == 1 (mod k).
- A104384 (program): Number of partitions of triangular numbers n*(n+1)/2 into (n-2) distinct parts for n>=3.
- A104385 (program): Number of distinct partitions of triangular numbers n*(n+1)/2 into 3 parts for n>=1.
- A104401 (program): a(n) = A104235(n)/4.
- A104402 (program): Matrix inverse of triangle A091491, read by rows.
- A104403 (program): a(0)=0; for n>0, a(n) = A102371(4n)/4.
- A104406 (program): Number of numbers <= n having no 2 in ternary representation.
- A104407 (program): Number of Hamiltonian groups of order <= n.
- A104435 (program): Number of ways to split 1, 2, 3, …, 2n into 2 arithmetic progressions each with n terms.
- A104436 (program): Number of ways to split 1, 2, 3, …, 3n into 3 arithmetic progressions each with n terms.
- A104437 (program): Number of ways to split 1, 2, 3, …, 4n into 4 arithmetic progressions each with n terms.
- A104449 (program): Fibonacci sequence with initial values a(0) = 3 and a(1) = 1.
- A104454 (program): Expansion of 1/(sqrt(1-5x)*sqrt(1-9x)).
- A104455 (program): Expansion of e.g.f. exp(5*x)*(BesselI(0,2*x) - BesselI(1,2*x)).
- A104457 (program): Decimal expansion of 1 + phi = phi^2 = (3 + sqrt(5))/2.
- A104458 (program): Define the first two terms to be 2 and 3. All the other terms are obtained by concatenating the two previous terms.
- A104459 (program): Possible differences between adjacent palindromes.
- A104461 (program): Number of instances of nonprimes m in Pythagorean triples x,y,z such that x^2 + y^2 = z^2. Except for 1, the number of instances of composite numbers m in Pythagorean triples.
- A104462 (program): Convert the binary strings in A101305 to decimal.
- A104463 (program): Complement of {A072756(n): n>=2}.
- A104470 (program): Tribonacci equivalent of mousetrap sequence (A002467).
- A104471 (program): Triangle of degree numbers for certain polynomials.
- A104472 (program): Triangle of degree numbers for certain polynomials.
- A104473 (program): a(n) = binomial(n+2,2)*binomial(n+6,2).
- A104474 (program): a(n) = binomial(n+3,3)*binomial(n+7,3).
- A104475 (program): a(n) = binomial(n+4,4) * binomial(n+8,4).
- A104476 (program): a(n) = binomial(n+7,7)*binomial(n+11,7).
- A104477 (program): Number of successive squarefree intervals between primes.
- A104478 (program): a(n) = binomial(n+8,8)*binomial(n+12,8).
- A104481 (program): Bisection of A104477.
- A104487 (program): a(n+3) = 6a(n+2) - 10a(n+1) + 3a(n); a(0) = 1, a(1) = 4, a(2) = 14.
- A104489 (program): Read central column in Table 3 in reference from right to left, convert to base 10.
- A104490 (program): Read central column in Table 3 in reference from right to left.
- A104492 (program): Cube excess of the n-th prime.
- A104493 (program): Numbers n for which the cube excess of the n-th prime is prime.
- A104496 (program): Expansion of 2*(2*x+1)/((x+1)*(sqrt(4*x+1)+1)).
- A104497 (program): Expansion of sqrt(1-8x)/sqrt(1-4x).
- A104498 (program): Expansion of (1/2)*(1-sqrt(1-8*x)/sqrt(1-4*x)).
- A104502 (program): Number of partitions where no part is a multiple of 9.
- A104505 (program): Triangle, read by rows, equal to the right-hand side of the triangle A084610, with row n listing the coefficients of (1+x-x^2)^n: T(n,k) = [x^(n+k)] (1+x-x^2)^n, for n>=k>=0.
- A104506 (program): Column 1 of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1 + x - x^2)^n.
- A104507 (program): Row sums of triangle A104505, which is equal to the right-hand side of the triangle A084610 of coefficients in (1+x-x^2)^n.
- A104512 (program): a(n) is the minimum number that is the first of k > 1 consecutive integers whose sum equals n, or 0 if impossible.
- A104513 (program): The number of consecutive integers > 1 beginning with A104512(n), the sum of which equals n, or 0 if impossible.
- A104514 (program): a(n) = least number k > 1 of consecutive integers which sum to 2*n; or a(n) = 0 if n is a power of 2.
- A104521 (program): Fixed point of the morphism 0->{1}, 1->{1,0,1}.
- A104522 (program): Expansion of (-1+x+3*x^2-x^3)/((x+1)(3*x-1)(x-1)^2).
- A104523 (program): Numbers that are neither Fibonacci nor Lucas numbers.
- A104530 (program): Expansion of (1+sqrt(1-4x))/(4sqrt(1-4x)-2).
- A104531 (program): Expansion of (1+sqrt(1-4*x))/(5*sqrt(1-4*x)-3).
- A104532 (program): Expansion of (1+sqrt(1-4*x))/(6*sqrt(1-4*x)-4).
- A104533 (program): E.g.f.: exp(2x/(1-2x)).
- A104537 (program): Expansion of g.f.: (1+x)/(1+2*x+4x^2).
- A104538 (program): Expansion of (1 + 2*x) / (1 + 2*x + 4*x^2).
- A104545 (program): Number of Motzkin paths of length n having no consecutive (1,0) steps.
- A104547 (program): Number of Schroeder paths of length 2n having no UHD, UHHD, UHHHD, …, where U=(1,1),D=(1,-1), H=(2,0).
- A104548 (program): Triangle read by rows giving coefficients of Bessel polynomial p_n(x).
- A104550 (program): Number of horizontal segments in all Schroeder paths of length 2n (a horizontal segment is a maximal string of horizontal steps).
- A104551 (program): Expansion of x/((1-x)*sqrt(1+4*x^2)).
- A104553 (program): Sum of trapezoid weights of all Schroeder paths of length 2n.
- A104554 (program): Expansion of x(1-x)/(1-x+2x^3-x^4).
- A104555 (program): Expansion of x(1 - x)/(1 - x + x^2)^3.
- A104556 (program): Matrix inverse of triangle A001497 of Bessel polynomials, read by rows; essentially the same as triangle A096713 of modified Hermite polynomials.
- A104557 (program): Triangle T, read by rows, such that the unsigned columns of the matrix inverse when read downwards equals the rows of T read backwards, with T(n,n)=1 and T(n,n-1) = floor((n+1)/2)*floor((n+2)/2).
- A104559 (program): Triangle, read by rows, of the number of left factors of peakless Motzkin paths of length n having k number of U’s and D’s (i.e., number of paths from (0,0) to the line x=n, consisting of steps U=(1,1), H=(1,0), D=(1,1), that never go below the x-axis and a U step is never followed by a D step).
- A104562 (program): Inverse of the Motzkin triangle A064189.
- A104563 (program): A floretion-generated sequence relating to centered square numbers.
- A104565 (program): Reversion of Pell numbers A000129(n+1).
- A104566 (program): Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product H*R of the infinite lower triangular matrices H = [1; 1,2; 1,2,1; 1 2,1,2; …] and R = [1; 1,1; 1,1,1; 1,1,1,1; …].
- A104567 (program): Triangle read by rows: T(i,j) = i-j+1 if j is odd; T(i,j) = 2(i-j+1) if j is even (1 <= j <= i).
- A104568 (program): Triangle of numbers that are 0 or 1 mod 3.
- A104569 (program): Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product Q*R of the infinite lower triangular matrices Q = [1; 1,3; 1,3,1; 1 3,1,3; …] and R = [1; 1,1; 1,1,1; 1,1,1,1; …].
- A104570 (program): Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product R*Q of the infinite lower triangular matrices R = [1; 1,1; 1,1,1; 1,1,1,1; …] and Q = [1; 1,3; 1,3,1; 1,3,1,3; …].
- A104571 (program): Triangle T(n,k) = A042948(n-k+1) read by rows, 0<=k<=n.
- A104572 (program): Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product A*B of the infinite lower triangular matrices A = [1; 3, 1; 5, 3, 1; 7, 5, 3, 1; …] and B=[1; 2,1; 1,2,1; 2,1,2,1; …].
- A104574 (program): Sum of trapezoid weights of all Motzkin paths of length n.
- A104578 (program): A Padovan convolution triangle.
- A104581 (program): Expansion of 1/(1 + x + x^3 + x^4).
- A104582 (program): Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product of the lower triangular matrix (Fibonacci(i-j+1)) and of the lower triangular matrix all of whose entries are equal to 1 (for j <= i).
- A104583 (program): Triangle read by rows: T(i,j) is the (i,j)-entry (1 <= j <= i) of the product A*B of the matrices A = [1; 3,1; 5,3,1; 7,5,3,1; …]; B = [1; 1,2; 1,2,1; 1,2,1,2; …] (both infinite lower triangular matrices).
- A104584 (program): a(n) = (1/2) * ( 3*n^2 + n*(-1)^n ).
- A104585 (program): a(n) = (1/2) * ( 3*n^2 - n*(-1)^n ).
- A104586 (program): Pentagonal wave sequence triangle.
- A104587 (program): Triangle read by rows, given by the matrix product A * B where A (A094727) = [1; 2, 3; 3, 4, 5; 4, 5, 6, 7; …] and B = [1; 1, 1; 1, 1, 1; …] (both are infinite lower triangular matrices with the other terms zero).
- A104588 (program): Product of primes less than or equal to sqrt(n).
- A104589 (program): a(1)=1. a(n) = a(n-1) + (sum of terms, from among terms a(1) through a(n-1), which are prime or 1).
- A104594 (program): A129760/2.
- A104597 (program): Triangle T read by rows: inverse of Motzkin triangle A097609.
- A104598 (program): Expansion of (1-z-sqrt(1-4z))/(1-4z)^2.
- A104621 (program): Heptanacci-Lucas numbers.
- A104624 (program): Expansion of exp( arcsinh( -2*x ) ) in powers of x.
- A104625 (program): Expansion of 1/(sqrt(1-4*x) - x^2).
- A104626 (program): Numbers having three 1’s in their base-phi representation.
- A104629 (program): Expansion of (1-2*x-sqrt(1-4*x))/(x^2 * (1+2*x+sqrt(1-4*x))).
- A104630 (program): Expansion of x/(1-5*x+7*x^2-5*x^3+x^4).
- A104631 (program): Coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.
- A104632 (program): 1/n times A104631(n), the coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.
- A104633 (program): Triangle T(n,k) = k*(k-n-1)*(k-n-2)/2 read by rows, 1<=k<=n.
- A104634 (program): Triangle T(n,k) = (k-1-n)*(k-2-n)*(k+2*n)/6, 1<=k<=n.
- A104635 (program): Odd n such that 2*n+1 is prime.
- A104636 (program): Even n such that 2n+1 is prime.
- A104637 (program): Number of even digits in n-th prime.
- A104638 (program): Number of odd digits in n-th prime.
- A104639 (program): Number of even digits in n^3.
- A104640 (program): Number of odd digits in n^3.
- A104643 (program): Number of arrangements that can be formed by taking n distinct things out of 25.
- A104647 (program): a(n) = a(n-1) mod n + a(n-2) mod n; a(0) = 0, a(1) = 1.
- A104653 (program): Number of topologically distinct trees with n vertices, including Steiner trees.
- A104670 (program): a(n) = binomial(n+2, 2)*binomial(n+7, n).
- A104671 (program): a(n) = C(n+3,3)*C(n+8,n+0).
- A104672 (program): a(n) = C(n+4,4)*C(n+9,n+0).
- A104673 (program): a(n) = C(n+5,5)*C(n+10,n+0).
- A104674 (program): a(n) = binomial(n+6, 6) * binomial(n+11, n).
- A104675 (program): a(n) = C(n+1,n) * C(n+6,1).
- A104676 (program): a(n) = binomial(n+2,2) * binomial(n+7,2).
- A104677 (program): a(n) = binomial(n+3,3)*binomial(n+8,3).
- A104678 (program): a(n) = binomial(n+4,4) * binomial(n+9,4).
- A104679 (program): a(n) = C(n+5,5)*C(n+10,5).
- A104680 (program): a(n) = binomial(n+7,7)*binomial(n+12,7).
- A104682 (program): a(n) = Sum_{j=0..14} n^j.
- A104683 (program): Interlaces “2*n^2 - 1 is a square” with NSW numbers.
- A104684 (program): Triangle read by rows: T(n,k) is the number of lattice paths from (0,0) to (n,n) using steps E=(1,0), N=(0,1) and D=(1,1) (i.e., bilateral Schroeder paths), having k D=(1,1) steps.
- A104686 (program): a(n) = n*(n+1)/2 (mod 6).
- A104688 (program): Binomial transform of Moebius sequence.
- A104696 (program): Rearrangement of positive integers: change odd digits d to 10-d.
- A104697 (program): Rearrangement of positive integers: change even digits d to 10-d.
- A104698 (program): Triangle read by rows: T(n,k) = Sum_{j=0..n-k} binomial(k, j)*binomial(n-j+1, k+1).
- A104706 (program): First terms in the rearrangements of integer numbers (see comments).
- A104708 (program): Product of number of involutions on n letters and number of partitions of n
- A104709 (program): Triangle read by rows: T(n,k) = Sum_{j=0..n} 2^(n-j)*binomial(j,k) for n >= 0 and 0 <= k <= n; also, Riordan array (1/((1-x)*(1-2*x)), x/(1-x)).
- A104712 (program): Pascal’s triangle, with the first two columns removed.
- A104713 (program): Triangle T(n,k) = binomial(n,k), read by rows, 3 <= k <=n .
- A104714 (program): Greatest common divisor of a Fibonacci number and its index.
- A104715 (program): Triangle T(n,k) = (2*k-1)*A000217(n-k+1) read by rows, 1<=k<=n.
- A104716 (program): Triangle T(n,k) = (2k-3+4n)*(k-1-n)*(k-2-n)/6, 1<=k<=n.
- A104717 (program): First terms in the rearrangements of integer numbers (see comments).
- A104720 (program): Expansion of 1/((1-x)(1-x^2)(1-10x)).
- A104721 (program): Expansion of (1+x)^2/(1-4*x^2).
- A104722 (program): Self-convolution of repeated Catalan numbers.
- A104726 (program): Triangle generated as the matrix product of A026729 and A000012 (triangular views), read by rows.
- A104727 (program): Triangle T(n,k) = (k-1-n)*(k-2-n)*(k^2+k+2*k*n+3*n^2+5*n)/24 read by rows, 1<=k<=n.
- A104728 (program): Triangle T(n,k) = (k-1-n)*(k-2-n)*(k-2+2*n)/2 read by rows, 1<=k<=n.
- A104730 (program): Triangle read by rows: T(n,k)=C(n+1,k)-C(k,n-k+1).
- A104731 (program): Triangle T(n,k) = sum_{j=k..n} (j+1)*binomial(k,j-k), read by rows, 0<=k<=n.
- A104732 (program): Square array T[i,j]=T[i-1,j]+T[i-1,j-1], T[1,j]=j, T[i,1]=1, read by antidiagonals.
- A104734 (program): Triangle T(n,k) = sum_{j=k..n} (2n-2j+1)*binomial(k,j-k), read by rows, 0<=k<=n.
- A104738 (program): Positions of records in A104706.
- A104739 (program): Positions of records in A104717.
- A104740 (program): a(1) = 1; for n > 1: if n is even, a(n) = least k > 0 such that sum(i=1,n/2,a(2*i-1))/sum(j=1,n,a(j))>=1/4, or 1 if there is no such k; if n is odd, a(n) = largest k > 0 such that sum(i=1,(n+1)/2,a(2*i-1))/sum(j=1,n,a(j))<=1/3, or 1 if there is no such k.
- A104743 (program): Numbers m = n + 3^n such that the equation x = 3^(m-x) has solution x = 3^n.
- A104745 (program): a(n) = 5^n + n.
- A104746 (program): Array T(n,k) read by antidiagonals: T(1,k) = 2^k-1 and recursively T(n,k) = T(n-1,k) + A000337(k-1), n,k >= 1.
- A104747 (program): a(n) = (n-3)*2^n + n*(n+3)/2 + 3.
- A104762 (program): Triangle read by rows: row n contains first n nonzero Fibonacci numbers in decreasing order.
- A104763 (program): Triangle read by rows: Fibonacci(1), Fibonacci(2), …, Fibonacci(n) in row n.
- A104764 (program): Triangle T(n,k) = Lucas(n-k+1) read by rows, 1<=k<=n.
- A104765 (program): Triangle T(n,k) read by rows: row n contains the first n Lucas numbers A000204.
- A104766 (program): Triangle T(n,k) = A001629(n-k+2) read by rows, 1<=k<=n.
- A104767 (program): a(n)=n for n <= 3, a(n) = 2a(n-1) - 2a(n-2) + 2a(n-3) for n >= 4.
- A104768 (program): Number of matrices G with entries in Z satisfying G^2=G+1 and having the form 2G=[1+p q-2n | q+2n 1-p].
- A104769 (program): G.f. -x/(1+x-x^3).
- A104770 (program): G.f. (1+x^2)/(1+x-x^3).
- A104771 (program): G.f. (1-x+x^2)/(1+x-x^3).
- A104777 (program): Integer squares congruent to 1 mod 6.
- A104792 (program): Triangle T(n,k) = A000330(n-k), n>=1, 0<=k<n, read by rows.
- A104793 (program): Triangle T(n,k) = A023537(n-k), n >= 1, 0 <= k < n, read by rows.
- A104794 (program): Expansion of theta_4(q)^2 in powers of q.
- A104795 (program): Triangle T(n,k) = C(n,k)+1 for k<n; T(n,k) = 1 for k=n, read by rows.
- A104796 (program): Triangle read by rows: T(n,k) = (n+1-k)*Fibonacci(n+2-k), for n>=1, 1<=k<=n.
- A104797 (program): Triangle T(n,k) = Fib(n-k+4)-n-k-3, n>=1, 0<=k<n, read by rows.
- A104798 (program): Triangle T(n,k) = k*[Fib(n-k+3) - 1], read by rows.
- A104806 (program): “Round of hypotenuse”, see comments.
- A104854 (program): Number of n-digit numbers using digits 0 to n-1 each exactly once and containing no 3-digit sequence in increasing or decreasing order.
- A104855 (program): Triangle read by rows: T(n,k) (0 <= k <= n) is the number of three-dimensional lattice walks consisting of n unit steps, each in one of the six coordinate directions, starting at the origin, never going below the horizontal plane and having k vertical steps.
- A104857 (program): Positive integers that cannot be represented as the sum of distinct Lucas 3-step numbers (A001644).
- A104858 (program): Partial sums of the little Schroeder numbers (A001003).
- A104859 (program): Partial sums of A001764.
- A104860 (program): Prime next to (n + n-th prime).
- A104861 (program): Number of compositions (ordered partitions) of the n-th prime into n positive integers.
- A104862 (program): First differences of A014292.
- A104872 (program): Diagonal sums of A004248.
- A104878 (program): A sum-of-powers number triangle.
- A104879 (program): Row sums of a sum-of-powers triangle.
- A104880 (program): Diagonal sums of a sum-of-powers triangle.
- A104881 (program): Triangle T(n,k) = Sum_{j=0..k} (n-k)^(k-j), read by rows.
- A104882 (program): Diagonal sums of number triangle A104881.
- A104887 (program): Triangle T(n,k) = (n-k+1)-th prime, read by rows.
- A104891 (program): a(0) = 0; a(n) = 5*a(n-1) + 5.
- A104895 (program): a(0)=0; thereafter a(2n) = -2*a(n), a(2n+1) = 2*a(n) - 1.
- A104896 (program): a(0) = 0; a(n) = 7*a(n-1) + 7.
- A104897 (program): Difference between (n+prime(n)) and next prime.
- A104934 (program): Expansion of (1-x)/(1 - 3*x - 2*x^2).
- A104954 (program): Decimal expansion of the area of the regular triangle with circumradius 1.
- A104955 (program): Decimal expansion of the area of the regular 5-gon (pentagon) of circumradius = 1.
- A104956 (program): Decimal expansion of the area of the regular hexagon with circumradius 1.
- A104967 (program): Matrix inverse of triangle A104219, read by rows, where A104219(n,k) equals the number of Schroeder paths of length 2n having k peaks at height 1.
- A104969 (program): Sum of squares of terms in rows of triangle A104967.
- A104970 (program): Sum of squares of terms in even-indexed rows of triangle A104967.
- A104974 (program): A Fredholm-Rueppel triangle.
- A104976 (program): Row sums of A104975.
- A104977 (program): Defining sequence for an inverse Fredholm-Rueppel triangle.
- A104978 (program): Triangle where g.f. satisfies: A(x,y) = 1 + x*A(x,y)^2 + x*y*A(x,y)^3, read by rows.
- A104979 (program): Semidiagonal sums of triangle A104978: a(n) = Sum_{k=0..[n/2]} A104978(n-k,n-2*k).
- A104981 (program): Column 1 of triangle A104980; also equals column 0 of triangle A104986, which equals the matrix logarithm of A104980.
- A104982 (program): Column 3 of triangle A104980, omitting leading zeros.
- A105020 (program): Array read by antidiagonals: row n (n >= 0) contains the numbers m^2-n^2, m >= n+1.
- A105022 (program): Entries in the n-th row of Pascal’s triangle that have the 2’s bit set in their binary expansion.
- A105023 (program): a(n) = A102370(n) - n. Or, 2*A103185(n).
- A105024 (program): a(n) = A102371(n) + n. Or, 2*A103745.
- A105031 (program): Binary equivalents of A103185.
- A105032 (program): Binary equivalents of A103745.
- A105033 (program): Read binary numbers downwards to the right.
- A105034 (program): Binary equivalents of A105033.
- A105036 (program): a(0) = 0, a(1) = 4, a(2) = 8, a(3) = 116, for n>3 a(n) = 26*a(n-2) - a(n-4) + 12.
- A105037 (program): a(0) = 0, a(1) = 4, a(2) = 6, a(3) = 98, for n>3 a(n) = 22*a(n-2) - a(n-4) + 10.
- A105038 (program): Nonnegative n such that 6*n^2 + 6*n + 1 is a square.
- A105040 (program): Nonnegative n such that 7*n^2 + 7*n + 1 is a square.
- A105042 (program): Numbers n such that 10n - 1 is prime.
- A105043 (program): Numbers n such that 100*n - 1 is prime.
- A105044 (program): Numbers n such that 1000*n - 1 is prime.
- A105045 (program): a(0)=0, a(1)=1, a(2)=2, a(3)=3, a(4)=12; for n > 4, a(n) = 8*a(n-2) - a(n-4) - 3.
- A105051 (program): Define a(1)=0, a(2)=0, a(3)=15, a(4)=111 then a(n)=254*a(n-2)+126-a(n-4) also sequence such that 7*(a(n)^2) + 7*a(n) + 1 = a square.
- A105057 (program): Numbers n such that 10000 * n - 1 is prime.
- A105058 (program): Expansion of g.f. (1+8*x-x^2)/((1+x)*(1-6*x+x^2)).
- A105059 (program): Numbers n such that 100000n - 1 is prime.
- A105060 (program): Triangle read by rows in which the n-th row consists of the first n nonzero terms of A033312.
- A105062 (program): Triangle read by rows, based on the morphism f: 1->2, 2->3, 3->4, 4->5, 5->6, 6->{6,6,10,7}, 7->8, 8->9, 9->10, 10->11, 11->12, 12->{12,12,5,1}. First row is 1. If current row is a,b,c,…, then the next row is a,b,c,…,f(a),f(b),f(c),…
- A105063 (program): Define a(1)=0, a(2)=0, a(3)=8, a(4)=24 and then a(n)=66*a(n-2)+32-a(n-4).
- A105064 (program): Triangle read by rows: a(n,m) =a(n-1,m)+(m-1)!*n: n<=m.
- A105065 (program): First entry of the vector v(n), where v(0) is the 2 by 2 column vector [0,1], v(n)=(M(n-1)^(n-1))v(n-1) and M(k) is the 2 x 2 matrix [[0,1],[1,k]].
- A105067 (program): a(n) = Sum_{j=0..11} n^j.
- A105070 (program): T(n,k) = 2^k*binomial(n,2k+1), where 0 <= k <= floor((n-1)/2, n >= 1.
- A105072 (program): Number of permutations on [n] whose local maxima are in ascending order.
- A105073 (program): Define a(1)=0, a(2)=2 then a(n) = 3*a(n-1) - a(n-2), a(n+1) = 3*a(n)-a(n-1) and a(n+2) = 3*a(n+1) - a(n) + 2.
- A105076 (program): Numbers k such that 60*k^2 + 60*k + 1 is a square.
- A105077 (program): G.f. -(x^3+5x+5)/((x^2-x+1)*(x+1)^2).
- A105081 (program): a(n) = 1 + A003188(n - 1), n >= 1.
- A105082 (program): Expansion of (5+4x)/(1-2x-x^2).
- A105083 (program): Trajectory of 1 under the morphism 1 -> 12, 2 -> 3, 3 -> 1.
- A105084 (program): Triangle read by rows: a[n, m] = a[n - 1, m] + binomial[n, m]; n <=m
- A105085 (program): Write the terms of A102370 in base 2, read by upward-sloping diagonals and convert to base 10.
- A105086 (program): Sum of the divisors of n minus the least nontrivial proper divisor of n.
- A105087 (program): Absolute difference between the sums of the left and right diagonals of ordered 2 X 2 prime squares.
- A105088 (program): Sum of the sides of ordered 2 X 2 prime squares.
- A105089 (program): Sum of the primes in ordered 3 X 3 prime squares.
- A105090 (program): Sum of the left diagonal in ordered 3 X 3 prime squares.
- A105091 (program): Sum of the right diagonal in ordered 3 X 3 prime squares.
- A105092 (program): Sum of the sides of ordered 2 prime sided prime triangles.
- A105094 (program): Expansion of 8 * (eta(q^2) / eta(q)^2)^8 in powers of q.
- A105095 (program): Expansion of 8*eta(2*tau)^8/eta(tau)^16 + eta(tau/2)^8/eta(tau)^16.
- A105100 (program): Sum of ordered 3 prime sided prime triangles.
- A105104 (program): Write A102370 in binary (A103542), read backwards, omit leading zeros, convert to base 10.
- A105106 (program): Numbers k such that the string k101 is prime.
- A105107 (program): Numbers n such that 10000n + 1001 is prime.
- A105110 (program): Direct matrix (non-recursive) content of -n to n+1 symmetry matrices.
- A105115 (program): Numbers k such that the decimal representation of 1/k is neither terminating nor purely repeating.
- A105125 (program): Triangle read by rows: T(n,k) = n^3 + k^3, n >= 0, 0 <= k <= n.
- A105126 (program): Primes of the form 16n+9.
- A105127 (program): Primes of the form 32n+17.
- A105128 (program): Primes of the form 64n+33.
- A105129 (program): Primes of the form 128n+65.
- A105130 (program): Primes of the form 256n+129.
- A105131 (program): Primes of the form 512n+257.
- A105132 (program): Primes of the form 1024n + 513.
- A105133 (program): Numbers n such that 8n + 5 is prime.
- A105134 (program): Numbers n such that 16n+9 is prime.
- A105135 (program): Numbers n such that 32n+17 is prime.
- A105136 (program): Numbers n such that 64n+33 is prime.
- A105137 (program): Numbers n such that 128n+65 is prime.
- A105138 (program): Numbers n such that 256n+129 is prime.
- A105139 (program): Numbers n such that 512n+257 is prime.
- A105140 (program): Numbers n such that 1024n+513 is prime.
- A105145 (program): Numbers n such that the string n10001 is prime.
- A105146 (program): Numbers k such that the string k100001 is prime.
- A105149 (program): Number of even semiprimes k such that n^2 < k <= (n+1)^2.
- A105150 (program): Approximation to leading digit of n-th Fibonacci number.
- A105151 (program): Greatest numerator among the n! ratios equal to the continued fractions which have the permutations of (1,2,3,…,n) for terms.
- A105161 (program): Difference between n and the second-smallest prime larger than n.
- A105163 (program): a(n) = (n^3 - 7*n + 12)/6.
- A105174 (program): Numbers k such that k*(k+1)/4 - 1 and k*(k+1)/4 + 1 are twin primes.
- A105178 (program): Digits in order in which they appear in decimal expansion of e
- A105186 (program): Replace odd-positioned digits with 0 in ternary representation of n.
- A105187 (program): a(n) = determinant of the n X n matrix m(i,j)=(i+j+2)!/i!/j!.
- A105188 (program): a(n) = determinant of the n X n matrix m(i,j)=(i+j+3)!/i!/j!.
- A105198 (program): a(n) = n(n+1)/2 mod 4.
- A105199 (program): Decimal expansion of arctan(2).
- A105202 (program): Irregular triangle read by rows: row n gives the word f(f(f(…(1)))) [with n applications of f], where f is the morphism 1->{1,2,1}, 2->{2,3,2}, 3->{3,1,3}.
- A105203 (program): Trajectory of 1 under the morphism f: 1->{1,2,1}, 2->{2,3,2}, 3->{3,1,3}.
- A105206 (program): Number of edges in a pancyclic graph on n+2 vertices with the fewest possible edges.
- A105209 (program): Nearest integer to the cube root of n.
- A105216 (program): Maximum denominator among the n! ratios equal to the continued fractions which have the permutations of (1,2,3,…,n) for terms.
- A105217 (program): Let b(n) denote the Lucas numbers, A000032: a(n) = Sum{k=0..n}C(n,k)^2*(n-k)!*b(k).
- A105218 (program): a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*k^3.
- A105219 (program): a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*k^2.
- A105220 (program): Trajectory of 1 under the morphism 1->{1,2,1}, 2->{2,2,2}.
- A105221 (program): a(n) is the sum of n’s distinct prime factors below n.
- A105222 (program): Smallest integer m > 1 such that m^(n-1) == 1 (mod n).
- A105223 (program): Number of squares between prime(n) and 2*prime(n) inclusive.
- A105224 (program): Number of squares between n and 2*n inclusive.
- A105225 (program): a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = -1, a(2) = -2.
- A105228 (program): a(n) = A102370(n) + 1.
- A105234 (program): Central column of a Moebius-binomial triangle.
- A105235 (program): Partial sums of the central column of a Moebius-binomial triangle.
- A105236 (program): a(n+5) = (a(n+4)*a(n+1) + 2*a(n+3)*a(n+2))/a(n).
- A105244 (program): Functional substitution on {1,2,3}.
- A105249 (program): a(n) = binomial(n+2,n)*binomial(n+6,n).
- A105250 (program): a(n) = binomial(n+3,n)*binomial(n+7,n).
- A105251 (program): a(n) = binomial(n+4,n)*binomial(n+8,n).
- A105252 (program): a(n) = binomial(n+5,n)*binomial(n+9,n).
- A105253 (program): a(n) = binomial(n+6,n)*binomial(n+10,n).
- A105254 (program): a(n) = binomial(n+7,n)*binomial(n+11,n).
- A105260 (program): Triangle read by rows: T(n,k)=C(2n-2k,k), n>=0, 0<=k<=floor(2n/3).
- A105262 (program): a(n)=number of tilings of a 4 X n rectangle using tiles that are either 1 X 1 squares or trominoes (here by a tromino we mean a 2 X 2 square with the upper right 1 X 1 square removed; no rotations allowed).
- A105266 (program): a(1)=1 and, for n>1, a(n) is the smallest integer greater than a(n-1) such that no three terms x,y,z of the sequence, with x<y<z, satisfy z-y=y-x+1.
- A105277 (program): Let F(n) denote the Fibonacci numbers, A000045: a(n) = Sum_{k=0..n} C(n,k)^2*(n-k)!*F(k).
- A105278 (program): Triangle read by rows: T(n,k) = binomial(n,k)*(n-1)!/(k-1)!.
- A105279 (program): a(0)=0; a(n) = 10*a(n-1) + 10.
- A105280 (program): a(0)=0; a(n) = 11*a(n-1) + 11.
- A105281 (program): a(0)=0; a(n)=6*a(n-1)+6.
- A105283 (program): (2n)-th prime mod n.
- A105284 (program): a(n)/4^n is the measure of the subset of [0,1] remaining when all intervals of the form [b/2^m - 1/2^(2m+1), b/2^m + 1/2^(2m+1)] have been removed, with b and m positive integers, b<2^m and m<=n.
- A105291 (program): Triangle read by rows: T(m,n) = binomial(m!,n), m>=0, 0 <= n <= m!.
- A105292 (program): Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n, having leftmost column of height k.
- A105306 (program): Triangle read by rows: T(n,k) is the number of directed column-convex polyominoes of area n, having the top of the rightmost column at height k.
- A105309 (program): a(n) = |b(n)|^2 = x^2 + 3*y*2 where (x,y,y,y) is the quaternion b(n) of the sequence b of quaternions defined by b(0)=1,b(1)=1, b(n) = b(n-1) + b(n-2)*(0,c,c,c) where c = 1/sqrt(3).
- A105312 (program): a(n) = Sum_{j=0..15} n^j.
- A105314 (program): Write the natural numbers as an infinite sequence of digits, starting at the left; a(n) is the subset (i.e., the position in this sequence of the “counting digits”) of the first digit of the n-th square.
- A105320 (program): Digital expansion of Pi: numbers from each pair of successive digits.
- A105321 (program): Convolution of binomial(1,n) and Gould’s sequence A001316.
- A105332 (program): a(n) = n*(n+1)/2 mod 8.
- A105333 (program): a(n) = n*(n+1)/2 mod 16.
- A105334 (program): a(n) = n*(n+1)/2 mod 32.
- A105335 (program): a(n) = n*(n+1)/2 mod 64.
- A105336 (program): a(n) = n*(n+1)/2 mod 128.
- A105337 (program): a(n) = n*(n+1)/2 mod 256.
- A105338 (program): a(n) = n*(n+1)/2 mod 512.
- A105339 (program): a(n) = n*(n+1)/2 mod 1024.
- A105340 (program): a(n) = n*(n+1)/2 mod 2048.
- A105343 (program): Elements of even index in the sequence gives A005893, points on surface of tetrahedron: 2n^2 + 2 for n > 1.
- A105346 (program): 3-almost primes whose indices are 3-almost primes.
- A105348 (program): An indicator sequence for the Jacobsthal numbers.
- A105349 (program): Characteristic sequence for the Pell numbers.
- A105350 (program): Largest squared factorial dividing n!.
- A105352 (program): Numbers of points on successive rings of the simple square lattice.
- A105356 (program): Records in A105354.
- A105367 (program): Expansion of (1-x^3)/(1-x^5).
- A105368 (program): Expansion of (1-x-x^3+x^4)/(1-x^5).
- A105369 (program): Expansion of ((1+x)^3 - x^3)/((1+x)^5 - x^5).
- A105370 (program): Expansion of ((1+x)^4-(1+x)x^3)/((1+x)^5-x^5).
- A105371 (program): Expansion of (1-x)(1-x+x^2)/(1-3x+4x^2-2x^3+x^4).
- A105372 (program): Decimal expansion of Hypergeometric2F1[ -(1/4),3/4,1,1] = sqrt(Pi)/(Gamma[1/4]*Gamma[5/4]).
- A105374 (program): a(n) = 4*n^3 + 4*n.
- A105384 (program): Expansion of x/(1 + x + x^2 + x^3 + x^4).
- A105385 (program): Expansion of (1-x^2)/(1-x^5).
- A105392 (program): Frobenius number of the subsemigroup of the natural numbers generated by successive pairs of Lucas numbers.
- A105395 (program): A simple “Fractal Jump Sequence” (FJS).
- A105396 (program): A simple “Fractal Jump Sequence” (FJS).
- A105397 (program): Periodic with period 2: repeat [4,2].
- A105398 (program): A simple “Fractal Jump Sequence” (FJS).
- A105399 (program): Largest prime <= numbers of the form 6k+3 (duplicates removed).
- A105420 (program): Number of partitions of n into 3-smooth parts.
- A105422 (program): Triangle read by rows: T(n,k) is the number of compositions of n having exactly k parts equal to 1.
- A105423 (program): Number of compositions of n+2 having exactly two parts equal to 1.
- A105426 (program): a(0)=1, a(1)=5, a(n)=8*a(n-1)-a(n-2).
- A105427 (program): Numbers n such that the near-repdigit number consisting of a 1 followed by n 3’s (i.e., of form 1333…33) is composite.
- A105432 (program): Numbers n such that the near-repdigit number consisting of n-1 1’s followed by a terminal 3 (i.e., of the form 111…1113) is composite.
- A105438 (program): Triangle, row sums = (Fibonacci numbers - 2).
- A105441 (program): Numbers with at least two odd prime factors (not necessarily distinct).
- A105450 (program): a(n) = binomial(n+5,6) + binomial(n+3,3) + binomial(n+2,3) + binomial(n-1,1).
- A105452 (program): Numerator of (7 n -1)/3.
- A105454 (program): Numbers n such that n*prime(n)+(n+1)*prime(n+1) is prime.
- A105469 (program): Number of numbers of the form 6k+3 with prime(n) <= 6k+3 < prime(n+1).
- A105470 (program): a(n)=1 if there is number of the form 6k+3 with prime(n) <= 6k+3 <= prime(n+1), otherwise 0.
- A105471 (program): a(n) = Fibonacci(n) mod 100.
- A105472 (program): Next-to-last digit of n-th Fibonacci number in decimal representation, a(n) = 0 for n <= 6.
- A105475 (program): Triangle read by rows: T(n,k) is number of compositions of n into k parts when each even part can be of two kinds.
- A105476 (program): Number of compositions of n when each even part can be of two kinds.
- A105477 (program): Triangle read by rows: T(n,k) is the number of compositions of n into k parts when there are two kinds of part 2.
- A105479 (program): a(n) = C(n,2)*Bell(n-2) (cf. A000217, A000110).
- A105480 (program): Number of partitions of {1…n} containing 3 pairs of consecutive integers, where each pair is counted within a block and a string of more than 2 consecutive integers are counted two at a time.
- A105481 (program): Number of partitions of {1…n} containing 4 pairs of consecutive integers, where each pair is counted within a block and a string of more than 2 consecutive integers are counted two at a time.
- A105482 (program): Number of partitions of {1…n} containing 5 pairs of consecutive integers, where each pair is counted within a block and a string of more than 2 consecutive integers are counted two at a time.
- A105488 (program): Number of partitions of {1…n} containing 2 detached pairs of consecutive integers, i.e., partitions in which only 1- or 2-strings of consecutive integers can appear in a block and there are exactly two 2-strings.
- A105489 (program): Number of partitions of {1…n} containing 3 detached pairs of consecutive integers, i.e., partitions in which only 1- or 2-strings of consecutive integers can appear in a block and there are exactly three 2-strings.
- A105490 (program): Number of partitions of {1…n} containing 4 detached pairs of consecutive integers, i.e., partitions in which only 1- or 2-strings of consecutive integers can appear in a block and there are exactly four 2-strings.
- A105491 (program): Number of partitions of {1…n} containing 5 detached pairs of consecutive integers, i.e., partitions in which only 1- or 2-strings of consecutive integers can appear in a block and there are exactly five 2-strings.
- A105492 (program): Number of partitions of {1,…,n} containing 2 strings of 3 consecutive integers such that only v-strings of consecutive integers can appear in a block, where v = 1,2,3.
- A105493 (program): Number of partitions of {1,…,n} containing 3 strings of 3 consecutive integers such that only v-strings of consecutive integers can appear in a block, where v = 1,2,3.
- A105494 (program): Number of partitions of {1,…,n} containing 4 strings of 3 consecutive integers such that only v-strings of consecutive integers can appear in a block, where v = 1,2,3.
- A105495 (program): Triangle read by rows: T(n,k) is the number of compositions of n into k parts when parts equal to q are of q^2 kinds.
- A105498 (program): Trajectory of 1 under the morphism 1->{1,2}, 2->{1,4}, 3->{3,4}, 4->{3,4}.
- A105499 (program): Trajectory of 1 under the morphism 1->{2,1,2}, 2->{1,3,1}, 3->{3,2,3}.
- A105500 (program): Trajectory of 1 under the morphism 1->{1,2}, 2->{3,2}, 3->{3,4}, 4->{1,4}.
- A105501 (program): Numbers n such that 1 is the leading digit of the n-th Fibonacci number in decimal representation.
- A105502 (program): Numbers m such that 2 is the leading digit of the n-th Fibonacci number in decimal representation.
- A105503 (program): Numbers n such that 3 is the leading digit of the n-th Fibonacci number in decimal representation.
- A105504 (program): Numbers m such that 4 is the leading digit of the n-th Fibonacci number in decimal representation.
- A105505 (program): Numbers n such that 5 is the leading digit of the n-th Fibonacci number in decimal representation.
- A105506 (program): Numbers m such that 6 is the leading digit of the n-th Fibonacci number in decimal representation.
- A105507 (program): Numbers m such that 7 is the leading digit of the n-th Fibonacci number in decimal representation.
- A105508 (program): Numbers m such that 8 is the leading digit of the m-th Fibonacci number in decimal representation.
- A105509 (program): Numbers m such that 9 is the leading digit of the m-th Fibonacci number in decimal representation.
- A105511 (program): Number of times 1 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
- A105512 (program): Number of times 2 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
- A105513 (program): Number of times 3 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
- A105515 (program): Number of times 5 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
- A105518 (program): Number of times 8 is the leading digit of the first n+1 Fibonacci numbers in decimal representation.
- A105522 (program): Inverse of number triangle A105438.
- A105523 (program): Expansion of 1-x*c(-x^2) where c(x) is the g.f. of A000108.
- A105530 (program): Ternary modular Gray code for n.
- A105531 (program): Decimal expansion of arctan 1/3.
- A105532 (program): Decimal expansion of arctan(1/5).
- A105533 (program): Decimal expansion of arctan(1/7).
- A105553 (program): a(n) is the number of 1’s in A103528(n) written in base 2.
- A105555 (program): Let d = number of divisors of n; a(n) = d-th prime.
- A105559 (program): McKay-Thompson series of class 6E for the Monster group with a(0) = 3.
- A105560 (program): a(1) = 1, and for n >= 2, a(n) = prime(bigomega(n)), where prime(n) = A000040(n) and bigomega(n) = A001222(n).
- A105561 (program): a(n) is the m-th prime, where m is the number of distinct prime factors of n (A001221), a(1) = 1.
- A105562 (program): a(n) is the prime whose index is the greatest prime factor of n, for n >1; a(1)=2.
- A105563 (program): a(n) = if (exactly 4 Fibonacci numbers exist with exactly n digits) then 1, otherwise 0.
- A105564 (program): Number of blocks of exactly 4 Fibonacci numbers having equal length <= n.
- A105565 (program): a(n) = if (exactly 5 Fibonacci numbers exist with exactly n digits) then 1, otherwise 0.
- A105566 (program): Number of blocks of exactly 5 Fibonacci numbers having equal length <= n.
- A105570 (program): Nonsquarefree numbers in place: a(n) = n if n is not squarefree, 0 otherwise.
- A105571 (program): Numbers m such that m - 2 and m + 2 are semiprimes.
- A105574 (program): a(1) = 2; for n > 1, a(n) is the prime whose index is the least prime factor of n.
- A105575 (program): Largest primes < numbers of the form 6k (duplicates removed).
- A105576 (program): a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 3, a(1) = 4, a(2) = 0.
- A105577 (program): a(n+3) = 2*a(n+2) - 3*a(n+1) + 2*a(n); a(0) = 1, a(1) = 5, a(2) = 6.
- A105578 (program): a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 1, a(2) = 0.
- A105579 (program): a(n+3) = 2a(n+2) - 3a(n+1) + 2a(n); a(0) = 1, a(1) = 3, a(2) = 4.
- A105580 (program): a(n+3) = a(n) - a(n+1) - a(n+2); a(0) = -5, a(1) = 6, a(2) = 0.
- A105581 (program): Primes whose indices are palindromic.
- A105583 (program): Numbers k such that 101*k + 997 is prime.
- A105584 (program): Fixed point of the morphism 1 -> 34, 2 -> 32, 3 -> 12, 4 -> 14, starting from a(0) = 1.
- A105602 (program): Divide each Fibonacci number by its primitive part.
- A105603 (program): Sylvester-Jacobsthal cyclotomic numbers.
- A105604 (program): Sylvester dividends for Jacobsthal numbers.
- A105606 (program): Sylvester dividends for Pell numbers.
- A105610 (program): Numbers n such that both p1=2n+3 and p2=4n+5 are primes.
- A105611 (program): a(n) is the LCM of the Jacobsthal sequence {J(1),…,J(n)}.
- A105616 (program): Column 1 of triangle A105615.
- A105617 (program): Column 2 of triangle A105615.
- A105624 (program): Column 2 of triangle A105623.
- A105627 (program): Column 1 of triangle A105626.
- A105633 (program): Row sums of triangle A105632.
- A105635 (program): a(n) = (2*Pell(n+1) - (1+(-1)^n))/4.
- A105636 (program): Transform of n^3 by the Riordan array (1/(1-x^2), x).
- A105637 (program): a(n) = a(n-2)+a(n-3)-a(n-5).
- A105638 (program): Maximum number of intersections in self-intersecting n-gon.
- A105639 (program): Multiples of coefficients in an asymptotic series of Ramanujan.
- A105642 (program): Composite nonsquares and noncubes.
- A105644 (program): a(n) = floor((Pi+e)*n).
- A105660 (program): G.f. (1-x)(x^2-5x+3)/(x^4-6x^3+13x^2-6x+1).
- A105661 (program): a(n)=1 if n is a prime, 2 if n is an even semiprime, otherwise 0.
- A105670 (program): a(1)=1 then bracketing n by powers of 2 as f(t)=2^t for f(t) < n <= f(t+1), a(n) = f(t+1) - a(n-f(t)).
- A105671 (program): a(2n) = Lucas(2n+3)^2, a(2n+1) = Lucas(2n+1)^2.
- A105672 (program): a(1)=1, then bracketing n with powers of 3 as f(t)=3^t for f(t) < n <= f(t+1), a(n) = f(t+1) - a(n-f(t)).
- A105673 (program): One-half of theta series of square lattice (or half the number of ways of writing n > 0 as a sum of 2 squares), without the constant term, which is 1/2.
- A105674 (program): Highest minimal distance of any Type I (strictly) singly-even binary self-dual code of length 2n.
- A105676 (program): Highest minimal Hamming distance of any Type 3 ternary self-dual code of length 4n.
- A105678 (program): Highest minimal Hamming distance of any Type 4^H Hermitian linear self-dual code over GF(4) of length 2n.
- A105679 (program): Numbers k such that 997*k + 101 is prime.
- A105686 (program): Number of inequivalent codes attaining highest minimal Hamming distance of any Type 4^H Hermitian linear self-dual code over GF(4) of length 2n.
- A105693 (program): a(n) = Fibonacci(2n+2)-2^n.
- A105694 (program): 10^n-10^(n-2).
- A105695 (program): Expansion of (1-x)*c(x/(1+x)), where c(x) is the g.f. of the Catalan numbers (A000108).
- A105696 (program): Expansion of (1-x)/sqrt((1-3*x)/(1+x)).
- A105700 (program): a(n)=1 if n is a prime, 2 if n is a semiprime, otherwise 0.
- A105720 (program): Triangular matchstick numbers in the class of prime numbers: sum of n-th and next n primes.
- A105723 (program): a(n) = 3^n - (-1)^n.
- A105725 (program): Triangle read by rows: T(n,k)=(n+k)!/k! (0<=k<=n-1; n>=1).
- A105728 (program): Triangle read by rows: T(n,1) = 1, T(n,n) = n and for 1 < k < n: T(n,k) = T(n-1,k-1) + 2*T(n-1,k).
- A105734 (program): For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, with a(1)=1, a(2)=1.
- A105736 (program): For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1,a(2)=3.
- A105746 (program): a(n) = minimal c>0 such that (n+1)^2+4*n*c = d^2 is a square.
- A105747 (program): Number of ways to use the elements of {1,..,k}, 0<=k<=2n, once each to form a collection of n (possibly empty) lists, each of length at most 2.
- A105748 (program): Number of ways to use the elements of {1,..,k}, 0<=k<=2n, once each to form a collection of n (possibly empty) sets, each with at most 2 elements.
- A105749 (program): Number of ways to use the elements of {1,…,k}, 0 <= k <= 2n, once each to form a sequence of n sets, each having 1 or 2 elements.
- A105750 (program): RE(Product{k=0..n, 1+kI}), I=sqrt(-1).
- A105751 (program): Imaginary part of Product_{k=0..n} 1+k*I, I=sqrt(-1).
- A105752 (program): Expansion of e.g.f. cos(i*log(1 + x)), i = sqrt(-1).
- A105754 (program): Lucas 8-step numbers.
- A105755 (program): Lucas 9-step numbers.
- A105760 (program): Nonnegative numbers k such that 2k+7 is prime.
- A105770 (program): Expansion of (x^2-x+1)(4x^2+x+1) / ((1+x+x^2)(1-x)^3).
- A105772 (program): Numbers k such that 7*k + 2 is prime.
- A105773 (program): Numbers n such that 11*n + 97 is prime.
- A105775 (program): Numbers n such that 97*n + 11 is prime.
- A105783 (program): Number of terms among the first n primes that are divisors of the sum of the first n primes.
- A105785 (program): Number of different forests of rooted trees, without isolated vertices, on n labeled nodes.
- A105787 (program): a(1) = 1; a(m) = maximum numerator possible with a continued fraction [b(1);b(2),b(3),…b(m-1)], where (b(1),b(2),b(3),…b(m-1)) is a permutation of (a(1),a(2),a(3),…a(m-1)).
- A105792 (program): Largest prime <= numbers congruent (2,4) mod 6 (duplicates removed).
- A105795 (program): Shallow diagonal sums of the triangle k!*Stirling2(n,k): a(n) = Sum_{k=0..floor(n/2)} T(n-k,k) where T is A019538.
- A105796 (program): “Stirling-Bernoulli transform” of Jacobsthal numbers.
- A105800 (program): Greatest Fibonacci number that is a proper divisor of the n-th Fibonacci number; a(1) = a(2) = 1.
- A105801 (program): Fibonacci-Collatz sequence: a(1)=1, a(2)=2; for n > 2, let fib = a(n-1) + a(n-2); if fib is odd then a(n) = 3*fib + 1 else a(n) = fib/2.
- A105804 (program): a(n)=F(n-1)a(n-1)+F(n)a(n-2), where F = A000045 (Fibonacci numbers), a(0)=0, a(1)=1.
- A105809 (program): Riordan array (1/(1-x-x^2), x/(1-x)).
- A105810 (program): Inverse of a Fibonacci-Pascal matrix A105809.
- A105811 (program): Expansion of (1+x-x^2)/(1+x)^2.
- A105812 (program): Expansion of (1+x-x^2)/(1+x).
- A105814 (program): a(n) = n^2 + (n concatenated with n).
- A105824 (program): a(n) = sigma(n) mod 4.
- A105825 (program): a(n) = sigma(n) (mod 5).
- A105826 (program): a(n) = sigma(n) (mod 7).
- A105827 (program): a(n) = sigma(n) (mod 8).
- A105837 (program): Numbers n such that n^2 = 11*m^2 + 11*m + 1.
- A105838 (program): Nonnegative integers n such that 11*n^2 + 11*n + 1 is a square.
- A105840 (program): Primes of the form r(r(n)+1)+1, where A141468(n)=r(n)=n-th nonprime.
- A105844 (program): Numbers n such that 37*n^2 + 37*n + 1 is a square.
- A105849 (program): Row sums of number triangle A105848.
- A105851 (program): Binomial transform triangle, read by rows.
- A105852 (program): sigma(n) mod 9.
- A105853 (program): a(n) = sigma(n) (mod 10), i.e., unit’s digit of sigma(n).
- A105854 (program): Primes of the form 20*k + 3.
- A105855 (program): A104647(n+1) - A104647(n).
- A105856 (program): a(n) = a(n-1) + A104647(n), a(0) = 0.
- A105861 (program): a(n) = (n/2) * Sum_{k=0..n} binomial(n,k)/gcd(n,k).
- A105862 (program): a(n) = n * Sum_{d|n} binomial(n,d)/gcd(n,d).
- A105863 (program): a(n) = n * Sum_{d|n} (binomial(n,d) / gcd(n,d)).
- A105864 (program): Expansion of (1/(1-x^2))*c(x/(1-x^2)), where c(x) is the g.f. of A000108.
- A105865 (program): Expansion of (1/(1-2*x^2))*c(x/(1-2*x^2)), where c(x) is the g.f. of A000108.
- A105866 (program): A generalized Chebyshev transform of the Fibonacci numbers.
- A105867 (program): A generalized Chebyshev transform of the Jacobsthal numbers.
- A105868 (program): Triangle read by rows, T(n,k) = C(n,k)*C(k,n-k).
- A105870 (program): Fibonacci sequence (mod 7).
- A105871 (program): a(n) = sum{k=0..floor(n/2), C(2*n-3*k, n)*C(n-k, k)}
- A105872 (program): a(n) = Sum_{k=0..floor(n/2)} C(2n-3k, n).
- A105876 (program): Primes for which -4 is a primitive root.
- A105899 (program): Period 6: repeat [1, 1, 2, 2, 3, 3].
- A105926 (program): First differences of A000166.
- A105927 (program): Let d(n) = A000166(n); then a(n) = ( (n^2+n-1)*d(n) + (-1)^(n-1)*(n-1) )/2.
- A105928 (program): a(n) = ((n^3 - 4n + 1)*A000166(n) + (-1)^(n+1)*(n-1)^2) / 6.
- A105930 (program): Starting position of the n-th prime in the almost-natural numbers sequence A007376.
- A105931 (program): a(1) = 1 then a(n) = a(n-1) - (-1)^ceiling(n/2)*a(floor(n/2)).
- A105938 (program): a(n) = binomial(n+2,2)*binomial(n+5,2).
- A105939 (program): a(n) = binomial(n+3,3)*binomial(n+6,3).
- A105940 (program): a(n) = binomial(n+5, n)*binomial(n+8, 5).
- A105942 (program): a(n) = C(n+6,n)*C(n+9,6).
- A105943 (program): a(n) = C(n+7,n) * C(n+10,7).
- A105944 (program): a(n) = C(n+8,n)*C(n+11,8).
- A105946 (program): a(n) = C(n+5,n)*C(n+3,3).
- A105947 (program): a(n) = C(n+6,n)*C(n+4,4).
- A105948 (program): a(n) = C(n+7,n)*C(n+5,5).
- A105951 (program): a(2*n) = -(2^(2*n+1) + 1), a(2*n+1) = (2^(n+1) - (-1)^n)^2.
- A105952 (program): (2n)-th Legendre polynomial P_{2n}(x), evaluated at x = 2n-1. Here the Legendre polynomials are normalized so that P_{n}(1) = 1.
- A105954 (program): Array read by descending antidiagonals: A(n, k) = (n + 1)! * H(k, n + 1), where H(n, k) is a higher-order harmonic number, H(0, k) = 1/k and H(n, k) = Sum_{j=1..k} H(n-1, j)), for 0 <= k <= n.
- A105955 (program): a(n) = Fibonacci(n) mod 11.
- A105960 (program): Smallest integer q >= 1 such that difference between q*sqrt(2) and the nearest integer is <= 1/n.
- A105961 (program): Primes p such that 20*p + 3 is prime.
- A105963 (program): Expansion of (1+4*x)/(1-x-3*x^2).
- A105964 (program): Expansion of x*(1+x^3-x^6+x^7)/(1-x^6)^2.
- A105968 (program): a(n) = 4*a(n-1) - a(n-2) - 2*(-1)^n, a(0) = 1, a(1) = 4.
- A105994 (program): Fibonacci sequence (mod 13).
- A105995 (program): Fibonacci sequence (mod 14).
- A105997 (program): Semiprime function n -> A001358(n) applied three times to n.
- A105998 (program): Semiprime function n -> A001358(n) applied four times to n.
- A106002 (program): a(n)=1 if there is a number of the form 6k+3 such that prime(n) < 6k+3 < prime(n+1), otherwise 0.
- A106005 (program): Fibonacci sequence (mod 15).
- A106006 (program): [n/2] + [n/3] + [n/5].
- A106033 (program): a(n) is the least number k such that n*prime(n)+k is a perfect square.
- A106034 (program): a(n) is the least number such that n*prime(n)+a(n) is a perfect cube.
- A106035 (program): The “Octanacci” sequence: Trajectory of 1 under the morphism 1->{1,2,1}, 2->{1}.
- A106036 (program): Trajectory of 1 under the morphism 1->{1,2}, 2->{1,2,3}, 3->{1,2,3,3}.
- A106040 (program): First 9-free digit in the fractional part of the decimal expansion of (1/10^n)^(1/10^n).
- A106043 (program): First digit other than 9 in the fractional part of the decimal expansion of (1/1000^n)^(1/1000^n).
- A106044 (program): Difference between n-th prime and next larger perfect square.
- A106057 (program): Primes p such that 1*p + 4 and 4*p + 1 are primes.
- A106058 (program): 4th diagonal of triangle in A059317.
- A106092 (program): Even numbers and primes.
- A106093 (program): Primes with maximal digit = 9.
- A106101 (program): Primes with minimal digit = 1.
- A106103 (program): Primes with minimal digit = 3.
- A106108 (program): Rowland’s prime-generating sequence: a(1) = 7; for n > 1, a(n) = a(n-1) + gcd(n, a(n-1)).
- A106113 (program): 5th diagonal of triangle in A059317.
- A106118 (program): Primes with maximal digit > 1.
- A106119 (program): Primes with maximal digit > 2.
- A106120 (program): Primes with maximal digit > 3.
- A106121 (program): Primes with maximal digit > 4.
- A106122 (program): Primes with maximal digit > 5.
- A106123 (program): Primes with maximal digit > 6.
- A106124 (program): Primes with maximal digit > 7.
- A106137 (program): N-th semiprime mod n.
- A106138 (program): Semiprimes (mod 2).
- A106139 (program): Semiprimes (mod 3).
- A106140 (program): Semiprimes (mod 4).
- A106141 (program): Semiprimes (mod 5).
- A106142 (program): Semiprimes (mod 6).
- A106143 (program): Semiprimes (mod 7).
- A106144 (program): Semiprimes (mod 8).
- A106145 (program): Semiprimes (mod 9).
- A106146 (program): Semiprimes (mod 10).
- A106147 (program): Image of 1 under the repeated morphism 1 -> 21, 2 -> 32, 3 -> 43, 4 -> 14.
- A106149 (program): Number of prime factors with multiplicity of the difference between consecutive primes.
- A106151 (program): In binary representation of n: delete one zero in each contiguous block of zeros.
- A106154 (program): Generation 5 of the substitution 1->{2, 1, 2}, 2->{3, 2, 3}, 3->{4, 3, 4}, 4->{5, 4, 5}, 5->{6, 5, 6}, 6->{1, 6, 1}, starting with 1.
- A106157 (program): G.f. (1-x-x^3+x^4-2*x^2)/((1-2*x)*(x-1)^2*(x+1)^2).
- A106160 (program): Highest minimal Hamming distance of Hermitian Type IV self-dual codes over GF(2) X GF(2) and length 2n.
- A106174 (program): a(n) = 2*n*a(n-1) - a(n-2), with a(0)=0, a(1)=1.
- A106180 (program): Matrix inverse of number triangle A046854.
- A106181 (program): Expansion of c(-x^2)(1+2x-sqrt(1+4x^2))/2, c(x) the g.f. of A000108.
- A106183 (program): Expansion of 1/sqrt(1-4x-4x^2+16x^3).
- A106184 (program): Expansion of 1/sqrt(1-4*x-8*x^2+32*x^3).
- A106185 (program): Expansion of 1/sqrt(1-4*x-12*x^2+48*x^3).
- A106186 (program): Expansion of 1/sqrt(1-4x+4x^2-16x^3).
- A106187 (program): Sequence array for central binomial numbers A000984.
- A106188 (program): Expansion of 1/((1-x^2)*sqrt(1-4*x)).
- A106189 (program): Expansion of 1/((1-2x^2)sqrt(1-4x)).
- A106190 (program): Triangle read by rows: T(n,k) = binomial(2(n-k),n-k)/(1-2(n-k)).
- A106191 (program): Expansion of sqrt(1-4x)/(1-x).
- A106192 (program): Expansion of sqrt(1-4x)/(1-x^2).
- A106193 (program): Expansion of sqrt(1-4x)/(1-2x^2).
- A106194 (program): Triangle read by rows, generated from binomial transforms of odd numbers.
- A106195 (program): Riordan array (1/(1-2*x), x*(1-x)/(1-2*x)).
- A106197 (program): Analog of A094091 for S=4.
- A106198 (program): Triangle, columns = successive binomial transforms of Fibonacci numbers.
- A106201 (program): Expansion of Re(x/(1-x-2*i*x^2)), i=sqrt(-1).
- A106202 (program): Expansion of Im(x/(1 - x - 2*i*x^2)), i=sqrt(-1).
- A106206 (program): Coefficients of (1 + 144*x)^(1/24).
- A106228 (program): G.f. satisfies: A(x) = 1 + x*A(x)/(1 - x*A(x)^2).
- A106229 (program): Least j > 1 for n > 0 such that j^2 = (n^2 + 1)*(k^2) + (n^2 + 1)*k + 1 where k sequence = A106230.
- A106230 (program): Least k > 0 for n > 0 such that (n^2 + 1)*(k^2) + (n^2 + 1)*k + 1 = j^2 where j sequence = A106229.
- A106231 (program): Least j > 1 such that j^2 = (4*n^2 + 2)*(k^2) + (4*n^2 + 2)*k + 1.
- A106232 (program): Least k > 0 such that (4*n^2 + 2)*(k^2) + (4*n^2 + 2)*k + 1 = j^2.
- A106233 (program): An inverse Catalan transform of A003462.
- A106244 (program): Number of partitions into distinct prime powers.
- A106246 (program): Number triangle T(n,k)=C(n,k)C(2,n-k).
- A106247 (program): Expansion of (1+2*x-x^2-2*x^3+x^4) / (1-x^2)^3.
- A106248 (program): McKay-Thompson series of class 5B for the Monster group with a(0) = -6.
- A106249 (program): Expansion of (1-x+x^2+x^3)/(1-x-x^4+x^5).
- A106250 (program): Expansion of (1-x+x^2+x^3)/(1-x-x^5+x^6).
- A106251 (program): Expansion of (1-x+x^2+x^3+x^5)/(1-x-x^6+x^7).
- A106252 (program): Number of positive integer triples (x,y,z), with x<=y<=z<=n, such that each of x,y and z divides the sum of the other two.
- A106253 (program): First difference of A106252.
- A106255 (program): Triangle composed of triangular numbers, row sums = A006918.
- A106256 (program): Numbers n such that 12*n^2 + 13 is a square.
- A106257 (program): Numbers k such that k^2 = 12*n^2 + 13.
- A106258 (program): Expansion of 1/sqrt(1-8x-8x^2).
- A106259 (program): Expansion of 1/sqrt(1-12x-12x^2).
- A106260 (program): Expansion of 1/sqrt(1-16x-16x^2).
- A106261 (program): Expansion of 1/sqrt(1 - 20*x - 20*x^2).
- A106262 (program): An invertible triangle of remainders of 2^n.
- A106263 (program): Row sums of number triangle A106262.
- A106264 (program): Diagonal sums of number triangle A106262.
- A106268 (program): Number triangle T(n,k) = binomial(k-n, n-k)*(-1)^(n-k) = (0^(n-k) + binomial(2*(n-k), n-k))/2 if k <= n, 0 otherwise; Riordan array (1/(2-C(x)), x) where C(x) is g.f. for Catalan numbers A000108.
- A106269 (program): Expansion of 1/((1 - x^2)*(2 - c(x))), where c(x) is the g.f. of A000108.
- A106270 (program): Inverse of number triangle A106268; triangle T(n,k), 0 <= k <= n.
- A106271 (program): Row sums of number triangle A106270.
- A106272 (program): Antidiagonal sums of number triangle A106270.
- A106291 (program): Period of the Lucas sequence A000032 mod n.
- A106292 (program): Period of the Lucas sequence A000032 mod prime(n).
- A106314 (program): Triangle T(n,k) composed of the squares min(n,k)^2.
- A106315 (program): Harmonic residue of n.
- A106316 (program): Remainder of the harmonic residue of n when divided by n.
- A106317 (program): Numbers n such that the remainder of the harmonic residue of n when divided by n is n-1.
- A106318 (program): Bhaskara twins: n such that 2*n^2 = X^3 and 2*n^3 = Y^2.
- A106325 (program): Smallest semiprime not less than n.
- A106326 (program): Smallest odd semiprime not less than n.
- A106328 (program): Numbers j such that 8*(j^2) + 9 = k^2 for some positive number k.
- A106329 (program): Numbers k such that k^2 = 8*j^2 + 9.
- A106330 (program): Numbers k such that k^2 = 24*(j^2) + 25.
- A106331 (program): Numbers j such that 24*(j^2) + 25 = k^2.
- A106344 (program): Triangle read by rows: T(n,k) = binomial(k,n-k) mod 2.
- A106345 (program): Diagonal sums of number triangle A106344.
- A106347 (program): A generalized Fredholm-Rueppel sequence.
- A106348 (program): Partial sums of a generalized Fredholm-Rueppel sequence.
- A106349 (program): Primes indexed by semiprimes.
- A106350 (program): Semiprimes indexed by primes.
- A106352 (program): Number of compositions of n into 3 parts such that no two adjacent parts are equal.
- A106370 (program): Smallest b>1 such that n contains no zeros in its base b representation.
- A106387 (program): Numbers j such that 6j^2 + 6j + 1 = 11k.
- A106388 (program): Numbers k such that 11k = 6j^2 + 6j + 1.
- A106389 (program): Numbers j such that 6j^2 + 6j + 1 = 13k.
- A106390 (program): Numbers k such that 13k = 6j^2 + 6j + 1.
- A106391 (program): A “cosh transform” of binomial(2n,n-1).
- A106392 (program): Expansion of 1/(1 - 6*x + 10*x^2).
- A106393 (program): Expansion of 1/(1 - 8x + 17x^2).
- A106397 (program): Binomial transform of Mertens’s function sequence A002321.
- A106400 (program): Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 1 and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 1’s and -1’s.
- A106401 (program): Expansion of (eta(q) * eta(q^9))^3 / eta(q^3)^2 in powers of q.
- A106402 (program): Expansion of eta(q^3)^9 / eta(q)^3 in powers of q.
- A106404 (program): Number of even semiprimes dividing n.
- A106405 (program): Number of odd semiprimes dividing n.
- A106406 (program): Expansion of (eta(q) * eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q.
- A106407 (program): Expansion of x((1-x)(1-x^2)(1-x^4)(1-x^8)…)^2.
- A106408 (program): Triangle, read by rows, where T(1,1) = 1; T(2,1) = T(2,2) = 2; for n > 2, T(n,n) = T(n-1,n-1) + T(n-2,n-2); T(n+1,n) = 2 * T(n,n); for all other entries, T(n,k) = T(n-1,k) + T(n-2,k).
- A106409 (program): n XOR (greatest proper divisor of n).
- A106434 (program): The (1,1)-entry of the matrix A^n, where A = [0,1;2,3].
- A106435 (program): a(n) = 3*a(n-1) + 3*a(n-2), a(0)=0, a(1)=3.
- A106436 (program): Difference array of Bell numbers A000110 read by antidiagonals.
- A106438 (program): G.f.: x(2-5x-2x^2)/(1-6x+9x^2-x^4).
- A106440 (program): a(n) = binomial(2n+4,n)*binomial(n+4,4).
- A106448 (program): Table of (x+y)/gcd(x,y) where (x,y) runs through the pairs (1,1), (1,2), (2,1), (1,3), (2,2), (3,1), …
- A106450 (program): a(n) = A004443(n) if n is odd, a(n) = A004443(n)/2 if n is even.
- A106458 (program): Bernoulli number denominators, with zeros at the odd places.
- A106459 (program): Expansion of f(-x, -x^3) in powers of x where f(,) is Ramanujan’s general theta function.
- A106461 (program): Binomial transform of the Tower of Hanoi sequence.
- A106462 (program): Binomial transform of A007318 (Pascal’s triangle by rows).
- A106464 (program): Antidiagonal sums of number triangle A003989.
- A106465 (program): A number triangle of GCDs mod 2.
- A106466 (program): Interleave 1,2,3,.. with 1,1,2,2,3,3,…
- A106467 (program): Inverse of number triangle A106465.
- A106468 (program): Absolute value of inverse of number triangle A106465.
- A106469 (program): Expansion of (1+x^2)(1+2x)/(1-x^2).
- A106470 (program): Inverse of number triangle A106468.
- A106471 (program): A number triangle with duplicated columns of the form 2^n-sum{j=0..2k-1, C(n,j)}.
- A106472 (program): Expansion of (1 - x)^2*(1 + x) / (1 - 2*x)^2.
- A106473 (program): Rows of A003989 expressed as base 10 numbers.
- A106474 (program): A006579(4n+4)/4.
- A106475 (program): An alternating sum of greatest common divisors.
- A106476 (program): Sequence array of Euler phi function.
- A106477 (program): Diagonal sums of Euler phi function sequence array.
- A106478 (program): Inverse of sequence array for Euler phi function.
- A106479 (program): First column in inverse of Euler phi sequence matrix.
- A106480 (program): Row sums of inverse of sequence array for Euler phi function.
- A106481 (program): An Euler phi transform of 1/(1-x^2).
- A106483 (program): Primes p such that 2p^2 - 1 is also prime.
- A106486 (program): Number of edges in combinatorial game trees.
- A106487 (program): Number of leaves in combinatorial game trees.
- A106489 (program): Triangle read by rows: T(n,k) is the number of short bushes with n edges and having the leftmost leaf at height k (a short bush is an ordered tree with no nodes of outdegree 1).
- A106490 (program): Total number of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.
- A106492 (program): Total sum of bases and exponents in Quetian Superfactorization of n, excluding the unity-exponents at the tips of branches.
- A106496 (program): Binomial transform of a fractal structured sequence.
- A106505 (program): Ordered and uniqued length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle.
- A106507 (program): G.f.: Product_{k>0} (1-x^(2k-1))/(1-x^(2k)).
- A106509 (program): Riordan array ((1+x)/(1+x+x^2), x/(1+x)), read by rows.
- A106510 (program): Expansion of (1+x)^2/(1+x+x^2).
- A106511 (program): Expansion of (1+x)^2/((1 + x + x^2)*(1 + x - x^2)).
- A106512 (program): Array read by antidiagonals: a(n,k) = number of k-colorings of a circle of n nodes (n >= 1, k >= 1).
- A106513 (program): A Pell-Pascal matrix.
- A106514 (program): Expansion of (1-x)/((1-2*x)*(1-2*x-x^2)).
- A106515 (program): A Fibonacci-Pell convolution.
- A106516 (program): A Pascal-like triangle based on 3^n.
- A106517 (program): Convolution of Fibonacci(n-1) and 3^n.
- A106521 (program): Numbers m such that Sum_{k=0..10} (m+k)^2 is a square.
- A106523 (program): Diagonal sums of number triangle A106522.
- A106524 (program): Interleave A038573(n+1) and 2*A038573(n+1).
- A106525 (program): Values of x in x^2 - 49 = 2*y^2.
- A106534 (program): Sum array of Catalan numbers (A000108) read by upward antidiagonals.
- A106539 (program): a(1)=1, a(2)=1, a(n) = (n-1)*a(n-1) - (n-2)*a(n-2) - … - a(1) for n>=3.
- A106540 (program): a(n) = a(n-1) - 2*a(n-2) - 3*a(n-3) - … - (n-1)*a(1), with a(1) = a(2) = 1, a(3) = -1.
- A106541 (program): a(n) = a(n-1) - 2*a(n-2) - 3*a(n-3) - … - (n-1)*a(1), with a(1) = a(2) = 2, a(3) = -2.
- A106542 (program): a(n) = a(n-1) - 2*a(n-2) - 3*a(n-3) - … - (n-1)*a(1), a(1) = a(2) = 3, a(3) = -3.
- A106543 (program): Composite numbers that are not perfect powers.
- A106544 (program): Perfect squares n^2 which are not the sum of two primes (otherwise 0).
- A106545 (program): a(n) = n^2 if n^2 is the sum of two primes, otherwise a(n) = 0.
- A106546 (program): a(n) = n^2 if n^2 is the difference of two primes, otherwise a(n) = 0.
- A106549 (program): a(n) = -1 if 2*n-1 is a prime, 1 if 2*n-1 is a prime squared, or 0 otherwise.
- A106562 (program): Perfect squares which are not the sum of two primes.
- A106563 (program): Numbers n such that n^2 is not the sum of two primes.
- A106564 (program): Perfect squares which are not the difference of two primes.
- A106565 (program): a(n) = 5*a(n-1) + 5*a(n-2) with a(0) = 0, a(1) = 5.
- A106566 (program): Triangle T(n,k), 0 <= k <= n, read by rows, given by [0, 1, 1, 1, 1, 1, 1, 1, … ] DELTA [1, 0, 0, 0, 0, 0, 0, 0, … ] where DELTA is the operator defined in A084938.
- A106567 (program): a(n) = 5*a(n-1) + 4*a(n-2), with a(0) = 4, a(1) = 4.
- A106568 (program): Expansion of 4*x/(1 - 4*x - 4*x^2).
- A106569 (program): a(n) = 5*a(n-1) + 3*a(n-2), where a(0) = 0, a(1) = 3.
- A106570 (program): a(n) = 4*a(n-1) + 3*a(n-2), with a(0)=0, a(1)=3.
- A106571 (program): Indices n of perfect squares n^2 which are not the difference of two primes.
- A106573 (program): Perfect squares which are neither the sum nor the difference of two primes.
- A106574 (program): Indices n of perfect squares n^2 which are neither the sum nor the difference of two primes.
- A106576 (program): Period 20. Sequence gives last digit of A106157, starting from the first positive term.
- A106578 (program): First differences of indices of squarefree central binomial numbers.
- A106579 (program): Triangular array associated with Schroeder numbers: T(0,0) = 1, T(n,0) = 0 for n > 0; T(n,k) = 0 if k < n; T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-1,k).
- A106586 (program): Digit next to last in squares ending in 6.
- A106587 (program): Sum of n-th prime squared and n-th perfect square.
- A106588 (program): Difference between n-th prime squared and n-th perfect square.
- A106594 (program): a(n) = number of primitive solutions to 4n+1 = x^2 + y^2.
- A106602 (program): Number of primitive positive solutions to 8n+2=x^2+y^2.
- A106603 (program): a(n) = - 2*a(n-1) - 8*a(n-3), a(0) = 1, a(1) = 1, a(2) = -2.
- A106607 (program): Expansion of (1+t^3)^2/((1-t)*(1-t^2)^2*(1-t^4)).
- A106608 (program): a(n) = numerator of n/(n+7).
- A106609 (program): Numerator of n/(n+8).
- A106610 (program): Numerator of n/(n+9).
- A106611 (program): a(n) = numerator of n/(n+10).
- A106612 (program): a(n) = numerator of n/(n+11).
- A106614 (program): a(n) = numerator of n/(n+13).
- A106615 (program): a(n) = numerator of n/(n+14).
- A106616 (program): Numerator of n/(n+15).
- A106617 (program): Numerator of n/(n+16).
- A106618 (program): a(n) = numerator of n/(n+17).
- A106619 (program): a(n) = numerator of n/(n+18).
- A106620 (program): a(n) = numerator of n/(n+19).
- A106621 (program): a(n) = numerator of n/(n+20).
- A106622 (program): Primes of the form r(r(r(n)+1)+1)+1, where A141468(n)=r(n)=n-th nonprime.
- A106624 (program): Expansion of g.f.: (1 - x^2 + x^3)/((1-x^2)*(1-2*x^2)).
- A106627 (program): Product L(n)*L_4(n), where L(n) are Lucas numbers and L_4(n) are Lucas 4-step numbers.
- A106633 (program): Number of ways to express n as k+l*m, with k, l, m all in the range [0..n].
- A106637 (program): Accumulation of permutation sequence on three symbols such that the a[n+2]-2*a[n+1]-a[n]=0 overall.
- A106638 (program): 3-symbol substitution that gives a dragon fractal.
- A106640 (program): Row sums of A059346.
- A106641 (program): A four-symbol four-at-a-time substitution with an ordering change: q=0.
- A106642 (program): A four-symbol four-at-a-time substitution with an ordering change: q=1.
- A106647 (program): Replace even digits d of n with 1+d/2.
- A106648 (program): a(n) = 3*n^2 + 6*n + 8.
- A106649 (program): Replace each digit d (except the leading one) of n with 9-d.
- A106664 (program): Expansion of g.f.: (1-3*x+x^2)/((1-x)*(1+x)*(1-2*x+2*x^2)).
- A106665 (program): Alternate paper-folding (or alternate dragon curve) sequence.
- A106666 (program): Expansion of g.f. (1-4*x^2+2*x^3)/((1-x)*(1-2*x-2*x^2+2*x^3)).
- A106671 (program): a(n) = ( prime(n + 2) * prime(n) - prime(n + 1)^2 ) modulo 3.
- A106690 (program): Numbers k such that 11*k - 97 is prime.
- A106692 (program): Numbers k such that 97*k - 11 is prime.
- A106693 (program): 3 symbols taken seven at a time symmetrically.
- A106695 (program): Numbers k such that 101*k - 997 is prime.
- A106701 (program): a(n) = next-to-most-significant binary digit of n-th composite positive integer.
- A106706 (program): a(0) = 19; for n>0, successively subtract 5, subtract 3 and double.
- A106707 (program): First entry of the vector (M^n)v, where M is the 2 X 2 matrix [[0,-1],[1,4]] and v is the column vector [0,1].
- A106709 (program): Expansion of g.f. -2*x/(1 - 5*x + 2*x^2).
- A106710 (program): Number of words with n letters from an alphabet of size 26 with at least two equal consecutive letters.
- A106729 (program): Sum of two consecutive squares of Lucas numbers (A001254).
- A106731 (program): Expansion of -2*x/(1 - 4*x + 2*x^2).
- A106732 (program): Expansion of -3*x/(1 - 5*x + 3*x^2).
- A106734 (program): a(n) = n^3 - 7*n + 7.
- A106737 (program): a(n) = Sum_{k=0..n} ({binomial(n+k,n-k)*binomial(n,k)} mod 2).
- A106740 (program): Triangle read by rows: greatest common divisors of pairs of Fibonacci numbers greater than 1: T(n, k) = gcd(Fibonacci(n), Fibonacci(k)).
- A106742 (program): a(n) = a(a(a(a(a(n - a(n-1)))))) + a(n - a(n-2)) with a(1) = a(2) = 1.
- A106743 (program): a(n) = -1 iff n is prime, a(n) = 1 iff n is not squarefree, otherwise (n is nonprime and squarefree) a(n) = 0.
- A106744 (program): Given n shoelaces, each with two aglets; sequence gives number of aglet pairs that must be picked up to guarantee that the probability that no shoelace is left behind is > 1/2.
- A106747 (program): Replace each odd digit d of n with (d-1)/2 and each even digit d with d/2.
- A106749 (program): Define the morphism f: 1->113, 2->13, 3->2; sequence gives trajectory of 1 under f.
- A106750 (program): Define the “Fibonacci” morphism f: 1->12, 2->1 and let a(0) = 2; then a(n+1) = f(a(n)).
- A106753 (program): Discriminants, negated, of definite binary quadratic forms.
- A106754 (program): Primes p with digital sum equal to 11.
- A106755 (program): Primes p with digital sum equal to 13.
- A106757 (program): Primes with digit sum = 16.
- A106759 (program): Primes with digit sum = 19.
- A106760 (program): Primes with digit sum = 20.
- A106789 (program): Sum of two consecutive squares of Lucas 3-step numbers (A001644).
- A106791 (program): Sum of two consecutive squares of Lucas 4-step numbers (A073817).
- A106793 (program): Number of words (over an alphabet of size 26) of length n with all different letters.
- A106799 (program): Number of prime factors of n apart from 2 or 3, counted with multiplicity.
- A106803 (program): Expansion of x*(1-x)/(1-2*x-x^2+x^3).
- A106804 (program): Expansion of g.f.: x*(2 - 9*x - 4*x^2)/((1 - 5*x + x^2)*(1 - 5*x - x^2)).
- A106805 (program): Expansion of g.f.: 1/(1 - 2*x - x^2 + x^3).
- A106825 (program): Trajectory of 1 under the morphism 1->1222, 2->2111.
- A106826 (program): Trajectory of 1 under the morphism 1->{2,1}, 2->{2,3}, 3->{4,3}, 4->{4,1}.
- A106829 (program): Given n shoelaces, each with two aglets; sequence gives number of aglets that must be picked up to guarantee that the probability that no shoelace is left behind is > 1/2.
- A106830 (program): Numerator of Sum_{ primes p <= n} 1/p.
- A106831 (program): Define a triangle in which the entries are of the form +-1/(b!c!d!e!…), where the order of the factorials is important; read the triangle by rows and record and expand the denominators.
- A106832 (program): 4*n-2 and 6*n alternatively.
- A106833 (program): 3n and 2n, alternating.
- A106835 (program): 4 X 4 vector Markov sequence with characteristic polynomial x^4-10*x^3+25*x^2-4.
- A106836 (program): First differences of A060833 and (from a(2) onward) also of A091067 and A255068.
- A106837 (program): Numbers n such that both n and n+1 have odd part of form 4k+3.
- A106838 (program): Numbers n such that n, n+1 and n+2 have odd part of form 4k+3.
- A106839 (program): Numbers congruent to 11 mod 16.
- A106840 (program): Numbers n such that both n and n+1 have odd part of form 4k+1.
- A106841 (program): Numbers m such that m, m+1 and m+2 have odd part of form 4k+1.
- A106842 (program): (1 + n + n^2)^n.
- A106843 (program): Numbers of form 3^i * prime(j), i>=0, j>0.
- A106844 (program): Exponent of 2 in A093641(n).
- A106845 (program): n^2 * (n^3 + 2n^2 + 7n - 2) / 8.
- A106846 (program): Sum {k + l*m <= n} (k + l*m), with 0 <= k,l,m <= n.
- A106847 (program): a(n) = Sum {k + l*m <= n} (k + l*m), with 0 < k,l,m <= n.
- A106849 (program): Values of n for which A106848(n) = n-1.
- A106851 (program): Let M = {{0, 0, 0, 1}, {1, 4, 0, 0}, {0, 1, 0, 0}, {0, 0, 1, 4}}, v[1] = {0, 1, 1, 2}’, v[n]=M.v[n-1]; then a(n) = v[n][[1]]
- A106852 (program): Expansion of 1/(1-x*(1-3*x)).
- A106853 (program): Expansion of 1/(1 - x + 4*x^2).
- A106854 (program): Expansion of 1/(1-x*(1-5*x)).
- A106855 (program): Expansion of 1/(1-x^2(1-3x)).
- A106859 (program): Primes of the form 2x^2 + xy + 2y^2.
- A106863 (program): Primes of the form x^2+xy+5y^2.
- A106865 (program): Primes of the form 2x^2 + 2xy + 3y^2.
- A106927 (program): Primes of the form 2x^2+xy+8y^2, with x and y any integer.
- A106949 (program): Primes of the form 2x^2 + 9y^2.
- A106950 (program): Primes of the form x^2 + 18y^2.
- A106952 (program): Primes of the form 3x^2-3xy+7y^2, with x and y nonnegative.
- A107003 (program): Primes of the form 24n + 5.
- A107006 (program): Primes of the form 4x^2-4xy+7y^2, with x and y nonnegative.
- A107007 (program): Primes of the form 3*x^2+8*y^2.
- A107008 (program): Primes of the form x^2 + 24*y^2.
- A107015 (program): Number of even terms in Zeckendorf representation of n.
- A107016 (program): Number of odd terms in Zeckendorf representation of n.
- A107017 (program): Second largest term in Zeckendorf representation of n, a(n)=0 if n itself is a Fibonacci number.
- A107025 (program): Binomial transform of the expansion of 1/(1-x^5-x^6).
- A107026 (program): Row sums of inverse of Riordan array (1/(1+x),x/(1+x)^4).
- A107033 (program): Expansion of f(x, x) * f(x, -x^2) in powers of x where f(,) is a Ramanujan theta function.
- A107034 (program): Expansion of f(-x) * f(-x^4) in powers of x where f() is a Ramanujan theta function.
- A107035 (program): Expansion of q * (psi(q^4) / phi(-q))^2 in powers of q where phi(), psi() are Ramanujan theta functions.
- A107040 (program): Indices of squarefree Pell numbers.
- A107041 (program): First differences of indices of squarefree Pell numbers.
- A107042 (program): First differences of indices of squarefree Catalan numbers.
- A107044 (program): A symmetric factorial triangle, read by rows: T(n,k) = min(n,k)!.
- A107048 (program): Denominators of coefficients that satisfy: 2^n = Sum_{k=0..n} c(k)*x^k for n>=0, where c(k) = A107047(k)/a(k).
- A107050 (program): Denominators of coefficients that satisfy: 3^n = Sum_{k=0..n} c(k)*x^k for n>=0, where c(k) = A107049(k)/a(k).
- A107054 (program): Denominators of coefficients that satisfy: 5^n = Sum_{k=0..n} c(k)*x^k for n>=0, where c(k) = A107053(k)/a(k).
- A107056 (program): Matrix inverse of A103247, so that T(n,k) = C(n,k)*A010842(n-k), read by rows.
- A107058 (program): a(n) = smallest number m>0 such that prime(n)*prime(n+1)-m is a prime.
- A107060 (program): a(n) = smallest number m >=0 such that n*prime(n)-m is a prime.
- A107061 (program): a(n) = largest number m >0 such that n*prime(n)-a(n) is a prime.
- A107063 (program): Expansion of q^(-1/24) * (eta(q^2) * eta(q^3)^4) / (eta(q) * eta(q^6)^2) in powers of q.
- A107064 (program): Expansion of q^(-17/24) * (eta(q) * eta(q^6)^4) / (eta(q^2) * eta(q^3)^2) in powers of q.
- A107065 (program): Riordan array (1/(1-x),x(1+x+x^2+x^3)).
- A107066 (program): Expansion of 1/(1-2*x+x^5).
- A107068 (program): Expansion of 1/((1+x)^3-x^4).
- A107071 (program): Numbers n such that 1019*n + 1021 is prime.
- A107073 (program): Numbers n such that the string 35n is prime.
- A107075 (program): Centered square numbers that are also centered pentagonal numbers.
- A107078 (program): Whether n has non-unitary prime divisors.
- A107079 (program): Minimal number of squared primes in a squarefree gap of length n.
- A107080 (program): McKay-Thompson series of class 4A for the Monster group.
- A107103 (program): Column 1 of triangle A107102, which is the matrix inverse of A100862.
- A107104 (program): Absolute row sums of triangle A107102, which is the matrix inverse of A100862.
- A107105 (program): Triangle, read by rows, where T(n,k) = C(n,k)*(C(n,k) + 1)/2, n>=k>=0.
- A107114 (program): Two-digit numbers from the decimal expansion of Pi.
- A107115 (program): Three-digit numbers from the decimal expansion of Pi (version 3) – but see comments.
- A107116 (program): Three-digit numbers from the decimal expansion of Pi (version 1).
- A107117 (program): Three-digit numbers from the decimal expansion of Pi (version 2).
- A107118 (program): Numbers that are both centered triangular numbers (A005448) and centered hexagonal numbers (A003215).
- A107128 (program): Divide the even digits of n by 2!.
- A107130 (program): Replace each odd digit d of n with (d-1)/2.
- A107131 (program): A Motzkin related triangle.
- A107134 (program): Primes of the form x^2+28y^2.
- A107135 (program): Primes of the form 5x^2 + 6y^2.
- A107144 (program): Primes of the form 5x^2 + 8y^2.
- A107145 (program): Primes of the form x^2 + 40y^2.
- A107151 (program): Primes of the form 5x^2 + 9y^2.
- A107152 (program): Primes of the form x^2 + 45y^2.
- A107167 (program): Primes of the form 5x^2 + 12y^2.
- A107168 (program): Primes of the form 4x^2 + 15y^2.
- A107169 (program): Primes of the form 3x^2 + 20y^2.
- A107181 (program): Primes of the form 8x^2 + 9y^2.
- A107223 (program): Floor(Pi*Floor(n*Pi)).
- A107227 (program): Numbers having no odd terms in their Zeckendorf representation.
- A107228 (program): Numbers having no even terms in their Zeckendorf representation.
- A107230 (program): A number triangle of inverse Chebyshev transforms.
- A107231 (program): a(n) = C(n+2,2)*C(n,floor(n/2)).
- A107232 (program): Expansion of (1+x*c(x^2))^3/sqrt(1-4*x^2), c(x) the g.f. of A000108.
- A107233 (program): An inverse Chebyshev transform of n^3.
- A107239 (program): Sum of squares of tribonacci numbers (A000073).
- A107240 (program): Sum of squares of first n tribonacci numbers (A000213).
- A107241 (program): Sum of squares of first n tetranacci numbers (A000288).
- A107242 (program): Sum of squares of tetranacci numbers (A001630).
- A107243 (program): Sum of squares of pentanacci numbers (A001591).
- A107244 (program): Sum of squares of hexanacci numbers (A001592, Fibonacci 6-step numbers).
- A107245 (program): Sum of squares of heptanacci numbers (Fibonacci 7-step numbers).
- A107246 (program): Sum of squares of octanacci numbers (Fibonacci 8-step numbers).
- A107249 (program): A number triangle with repeated columns of binomial coefficients.
- A107253 (program): a(n) = n^4 - 15*n + 15.
- A107255 (program): a(n) = n^5 - 31*n + 31, with n*a(n) + n*( n - 1 )*31 = n^6.
- A107256 (program): a(n) = n^6 - 63*n + 63, with n*a(n) + n*(n-1)*63 = n^7.
- A107258 (program): Numbers not representable as Fibonacci(i) + triangular(j), i,j>=0.
- A107259 (program): Number of ways to represent n as Fibonacci(i) + triangular(j), i,j>=0.
- A107264 (program): Expansion of (1 - 3*x - sqrt((1-3*x)^2 - 4*3*x^2))/(2*3*x^2).
- A107265 (program): Expansion of (1-5*x-sqrt((1-5*x)^2-4*5*x^2))/(2*5*x^2).
- A107266 (program): Expansion of (1-6*x-sqrt((1-6*x)^2-4*6*x^2))/(2*6*x^2).
- A107267 (program): A square array of Motzkin related transforms, read by antidiagonals.
- A107268 (program): Sums of antidiagonals of A107267.
- A107279 (program): a(n) = 1 if n is an odd prime, a(n) = 2 if n is a nonzero even number, otherwise a(n) = 0.
- A107281 (program): a(0) = 1, a(1) = 1, a(2) = 2 and for n >= 1: a(n+1) = SORT[a(n) + a(n-1) + a(n-2)] where SORT places digits in ascending order and deletes 0’s.
- A107283 (program): E.g.f. exp(x)*(x^2+x+2)/(1-x).
- A107284 (program): a(n)/4^n is the measure of the subset of [0,1] remaining when all intervals of the form [b/2^m - 1/2^(2m), b/2^m + 1/2^(2m)] have been removed, with b and m positive integers, b < 2^m and m <= n.
- A107285 (program): 5*401*(10^n + 1).
- A107286 (program): a(0) = 0; for n>0, minimal prime factor of n, or 1 if n is 1 or a prime.
- A107289 (program): Numbers k such that the sum of digits of k^2 is a prime.
- A107293 (program): The (1,1)-entry of the matrix M^n, where M is the 5 X 5 matrix [[0,1,0,0,0],[0,0,1,0,0], [0,0,0,1,0], [0,0,0,0,1], [1,0,-1,1,1]].
- A107294 (program): GCD of (n + prime(n)) and (n + 1 + prime(n+1)).
- A107298 (program): a(n) = 3*a(n-1)-a(n-2)-2*a(n-3)+a(n-4), n>5.
- A107299 (program): a(n) = 4*a(n-1)-2*a(n-2)-3*a(n-3)+2*a(n-4), n>5.
- A107300 (program): a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) with a(0)=3, a(1)=2, a(3)=8.
- A107303 (program): Numbers k such that (3*k - 5) is prime.
- A107304 (program): Numbers k such that 5k - 7 is prime.
- A107305 (program): Numbers k such that 11*k - 13 is prime.
- A107306 (program): Numbers k such that (17*k - 19) is prime.
- A107307 (program): G.f. (1-x-2*x^2-x^3+x^4)/((x-1)^3*(6*x^2+2*x-1)).
- A107308 (program): Numbers k such that (29*k - 31) is prime.
- A107309 (program): Concatenation of twin primes in reverse order.
- A107316 (program): Floor(exp(n)/n).
- A107317 (program): Semiprimes of the form 2*(m^2 + m + 1) (implying that m^2 + m + 1 is a prime).
- A107319 (program): a(n) = C(n+8,8)*C(n+6,6).
- A107323 (program): If n-th prime is 6m-1, then a(n) = 6m+1. If n-th prime is 6m+1, then a(n) = 6m-1.
- A107324 (program): Floor(A063655(n)/2).
- A107325 (program): a(n) = ceiling(A063655(n)/2).
- A107330 (program): a(n) = 4*a(n-1)-a(n-2)-3*a(n-3)+a(n-4), n>5.
- A107332 (program): The (1,3)-entry of the matrix M^n, where M is the 5x5 matrix [[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1],[1,0,-1,1,1]].
- A107334 (program): G.f.: (3-4*x-3*x^2)/(1-2*x-3*x^2+2*x^3).
- A107345 (program): From the binary representation of n: binomial(number of zeros, number of blocks of contiguous zeros).
- A107346 (program): Differences between successive permutations of 1,2,3,4,5 regarded as decimal numbers arranged in increasing order.
- A107347 (program): Number of even semiprimes strictly between prime(n) and 2*prime(n).
- A107351 (program): Expansion of (1+x^3)/((1-x)^3*(1-x^2)^3*(1-x^3)).
- A107358 (program): Dying rabbits: a(n) = Fibonacci(n) for n <= 12; for n >= 13, a(n) = a(n-1) + a(n-2) - a(n-13).
- A107359 (program): A003754(n+1) - A003754(n).
- A107361 (program): G.f. 1/((3*x-1)*(x^2-x-1)).
- A107366 (program): Numbers k such that 101*k + 103 is prime.
- A107368 (program): a(n) = a(n-1) + a(n-2) - a(n-3) + a(n-5).
- A107369 (program): Numbers n such that 103*n + 101 is prime.
- A107371 (program): Numbers k such that 101*k - 103 is prime.
- A107372 (program): Numbers n such that 103*n - 101 is prime.
- A107373 (program): a(n) = (n/2)*binomial(n-1, floor((n-1)/2)) - 2^(n-2).
- A107375 (program): a(n) = 6*a(n-1)-6*a(n-3)-a(n-4).
- A107376 (program): a(n) = 7*a(n-1)-7*a(n-3)-a(n-4).
- A107377 (program): Expansion of x*(1-4*x-3*x^2)/(1-5*x+5*x^3+x^4).
- A107378 (program): Expansion of x*(1-3*x-2*x^2)/(1-4*x+4*x^3+x^4).
- A107382 (program): a(n) = 4*a(n-1)-4*a(n-3)-a(n-4).
- A107383 (program): a(n) = 2*a(n-2) + 2*a(n-3).
- A107384 (program): a(n)= a(n-1) +3*a(n-2) -3*a(n-4).
- A107385 (program): a(n) = a(n-1)+4*a(n-2)-4*a(n-4).
- A107386 (program): a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4), n>6.
- A107387 (program): Expansion of x*(1-2*x-x^2)/( (1-x)*(1+x)*(1-3*x+x^2)).
- A107388 (program): Expansion of x*(3*x-1)*(2*x-1) / ( (1-x)*(1+x)*(x^2-4*x+1) ).
- A107389 (program): Expansion of x*(1-6*x+7*x^2)/( (1-x)*(1+x)*(1-5*x+x^2)).
- A107391 (program): Expansion of sin(x)^2 * sinh(x)^2.
- A107392 (program): Number of (inequivalent) fuzzy subgroups of the direct sum of group of integers modulo p^n and group of integers modulo 2 for a prime p with (p,2) = 1. Z_{p^n} + Z_2.
- A107393 (program): a(n) = -1 if n is a prime, else a(n) = 1 if n is the sum of three odd primes, else a(n) = 2 if n is the sum of two primes, else a(n) = 0.
- A107394 (program): C(n+2,2)*C(n+4,2).
- A107395 (program): a(n) = C(n+4,4)*C(n+6,4).
- A107396 (program): a(n) = binomial(n+5, 5) * binomial(n+7, 5).
- A107397 (program): a(n) = binomial(n+6, 6) * binomial(n+8, 6).
- A107398 (program): a(n) = binomial(n+7, 7) * binomial(n+9, 7).
- A107399 (program): a(n) = C(n+8,8)*C(n+10,8).
- A107400 (program): Numbers k such that 107*k + 109 is prime.
- A107401 (program): a(n) = -a(n-1)+4*a(n-2)+4*a(n-3)-a(n-4)-a(n-5).
- A107403 (program): Expansion of e.g.f. 1/(1-3*sinh(x)).
- A107405 (program): Numbers n such that 109*n + 107 is prime.
- A107406 (program): Numbers n such that 107*n - 109 is prime.
- A107407 (program): Numbers n such that 109*n - 107 is prime.
- A107409 (program): Each term is sum of three previous terms mod 10.
- A107410 (program): Each term is sum of three previous terms mod 9.
- A107417 (program): a(n) = C(n+2,2)*C(n+5,5).
- A107418 (program): a(n) = C(n+3,3)*C(n+6,6).
- A107419 (program): a(n) = C(n+4,4)*C(n+7,7).
- A107420 (program): a(n) = C(n+5,5)*C(n+8,8).
- A107421 (program): a(n) = C(n+6,6)*C(n+9,9).
- A107422 (program): a(n) = binomial(n+7,7) * binomial(n+10,10).
- A107423 (program): Numbers k equal prime(n)*prime(n+1) such that k+1 is a square and k-1 is even semiprime.
- A107427 (program): Maximal number of simple triangular regions that can be formed by drawing n line segments in the Euclidean plane.
- A107430 (program): Triangle read by rows: row n is row n of Pascal’s triangle (A007318) sorted into increasing order.
- A107436 (program): a(n) = (a^5)(n-1) + a(n-a(n-1)) = a(a(a(a(a(n-1))))) + a(n-a(n-1)), a(1) = a(2) = 1.
- A107443 (program): G.f. (3*x^2+1)/((1-x)*(2*x^2+x+1)*(2*x^2-x+1)).
- A107444 (program): a(n) = C(n^3, n).
- A107446 (program): a(n) = binomial(n^4, n).
- A107450 (program): Additive persistence of the prime numbers.
- A107453 (program): 1 followed by repetitions of the period-4 sequence 1,1,1,2.
- A107454 (program): Number of nonisomorphic generalized Petersen graphs P(n,k) with girth 5 on 2n vertices for 1<=k<=Floor[(n-1)/2].
- A107458 (program): Expansion of g.f.: (1-x^2-x^3)/( (1+x)*(1-x-x^3) ).
- A107459 (program): Number of nonisomorphic bipartite generalized Petersen graphs P(2n,k) with girth 6 on 4n vertices for 1<=k<n.
- A107461 (program): Number of gap-free compositions of n into distinct parts, cf. A107428.
- A107463 (program): a(0)=0, a(n)=1 if n is 1 or is a prime, otherwise sum of prime factors of n with multiplicity.
- A107464 (program): Number of fuzzy subgroups of rank 3 cyclic group of order (p^n)*q*r where p, q and r are three distinct prime.
- A107471 (program): a(n) = 3*prime(n) - 2*prime(n+1).
- A107473 (program): Sum of numerator and denominator of product{p|n,p=primes} (1 -1/p).
- A107479 (program): a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7).
- A107480 (program): a(n) = a(n-1) + a(n-3) + a(n-4) + a(n-5) + a(n-7).
- A107487 (program): Ordered semiperimeters of Pythagorean triangles.
- A107490 (program): Coefficients of a certain theta series.
- A107491 (program): Coefficients of a certain theta series.
- A107492 (program): Coefficients of a certain theta series.
- A107493 (program): Coefficients of a certain theta series.
- A107495 (program): Coefficients of a certain theta series.
- A107496 (program): Coefficients of a certain theta series.
- A107505 (program): Theta series of quadratic form with Gram matrix [ 2, 1, 0, 1; 1, 4, 1, 0; 0, 1, 4, -2; 1, 0, -2, 8].
- A107576 (program): a(n)=perimeter of n-th triangle listed at A107572.
- A107583 (program): a(n) = 3^n - 3*n.
- A107584 (program): a(n) = 4^n - 4*n.
- A107585 (program): a(n) = 5^n - 5*n.
- A107587 (program): Number of Motzkin n-paths with an even number of up steps.
- A107597 (program): Antidiagonal sums of triangle A107105: a(n) = Sum_{k=0..n} A107105(n-k,k), where A107105(n,k) = C(n,k)*(C(n,k) + 1)/2.
- A107599 (program): a(n) = 0,1,2 (resp.) if (1/2)[prime(n-1)+prime(n+1)] is less than, equal to or greater than prime(n) (resp.).
- A107604 (program): Order of appearance of twos in the Fibonacci substitution :triangular in form.
- A107615 (program): Coefficient list length of Poincaré-like polynomials made from A047845, indices of 4*n+1 nonprimes as the m(i) exponents.
- A107616 (program): Triangle read by rows, generated from arithmetic sequences.
- A107620 (program): Primes multiplied alternately by 3 and 2.
- A107621 (program): Primes multiplied alternately by 2 and 3.
- A107622 (program): Primes plus alternately 2 and 3.
- A107623 (program): Primes plus alternately 3 and 2.
- A107635 (program): McKay-Thompson series of class 32a for the Monster group.
- A107638 (program): Order of appearance of ones in the Fibonacci substitution :triangular in form.
- A107643 (program): Numbers n such that the string 75n is prime.
- A107653 (program): Expansion of q / (chi(q) * chi(q^3))^6 in powers of q where chi() is a Ramanujan theta function.
- A107659 (program): a(n) = Sum_{k=0..n} 2^max(k, n-k).
- A107660 (program): Sum 3^max(k,n-k),k=0..n.
- A107661 (program): Array read by antidiagonals: T(n,m) = Sum m^max(k,n-k),k=0..n.
- A107663 (program): a(2n) = 2*4^n-1, a(2n+1) = (2^(n+1)+1)^2; interlaces A083420 with A028400.
- A107665 (program): Numbers with semiprime digits (digits 4, 6, 9 only).
- A107680 (program): Repeating k-th ternary repunit (A003462) 2^k times, k >= 0.
- A107684 (program): Union of sequences 2^k-1, 2^k and 2^k+1.
- A107706 (program): Quadratic recurrence a(n)=2a(n-1)^2+a(n-2), a(0)=a(1)=1.
- A107712 (program): a(n) = Product_{k=1..n} prime(k+n).
- A107713 (program): Convolution of 2^n*n! and n!.
- A107715 (program): Primes whose decimal representation contains only digits from the set {0,1,2,3}.
- A107716 (program): Inverse INVERT transform of triple factorial numbers (3*n-2)!!! (A007559).
- A107730 (program): Numbers n such that prime(n+1) has the same last digit as prime(n).
- A107731 (program): Row 7 of the array in A107735.
- A107732 (program): Column 1 of the array in A107735.
- A107733 (program): Column 2 of the array in A107735.
- A107737 (program): Numbers n such that, in prime decomposition of n, sum of all prime factors and their orders is prime.
- A107738 (program): Primes as a sum of prime factors and their orders in prime decomposition of some n.
- A107740 (program): Number of numbers m such that prime(n) = m + (digit sum of m).
- A107742 (program): G.f.: Product_{j>=1} Product_{i>=1} (1 + x^(i*j)).
- A107743 (program): Numbers m such that m+(digit sum of m) is a composite number.
- A107744 (program): Smallest prime factor of 6*n+1.
- A107745 (program): Smallest prime factor of 6*n-1.
- A107746 (program): Numbers n such that the least prime factor of 6*n+1 > the least prime factor of 6*n-1, A107744(n) > A107745(n).
- A107747 (program): Numbers n such that the least prime factor of 6*n+1 < the least prime factor of 6*n-1, A107744(n) < A107745(n).
- A107749 (program): OrdinaryUnitarySigma(n): If n = Product p_i^r_i then OUSigma(n) = Sigma(2^r_1)*UnitarySigma(n/2^r_1).
- A107750 (program): If n=0 then 0, else smallest number greater than its predecessor and having either more or fewer zeros in its binary representation.
- A107751 (program): a(n) = A107750(n+1) - A107750(n).
- A107755 (program): Numbers k such that Sum_{j=1..k} Catalan(j) == 0 (mod 3).
- A107756 (program): Numbers k such that Sum_{j=1..k} Catalan(j) == 1 (mod 3).
- A107757 (program): Numbers k such that Sum_{j=1..k} Catalan(j) == 2 (mod 3).
- A107758 (program): (+2)Sigma(n): If n = Product p_i^r_i then a(n) = Product (2 + Sum p_i^s_i, s_i=1 to r_i) = Product (1 + (p_i^(r_i+1)-1)/(p_i-1)), a(1) = 1.
- A107759 (program): a(n) = (+2)UnitarySigma(n): if n = Product p_i^r_i then a(n) = Product (2 + p_i^r_i).
- A107760 (program): Expansion of eta(q^3) * eta(q^2)^6 / (eta(q)^3 * eta(q^6)^2) in powers of q.
- A107767 (program): a(n) = (1 + 3^n - 2*3^(n/2))/4 if n is even, (1 + 3^n - 4*3^((n-1)/2))/4 if n odd.
- A107769 (program): a(n) = (A001333(n+1) - 2*A005409(floor((n+3)/2)) - 1) / 4.
- A107770 (program): Index of greater of twin primes in the primes.
- A107771 (program): Numbers n such that 2*n + 5 and 5*n + 2 are primes.
- A107782 (program): In binary representation of n: (number of zeros) minus (number of blocks of contiguous zeros).
- A107785 (program): Sequence obtained using characteristic polynomial that is Laplace transform of the tribonacci characteristic polynomial: -s^4*L(t^3 -t^2 -t -1) = s^3 +s^2 +2*s -6.
- A107786 (program): a(n) = |b(n)| where b(n) = -b(n-1) + 6*b(n-3) with b(0)=0, b(1)=1, b(2)=1.
- A107789 (program): Trajectory of 2 under evenly many applications of the morphism 1 -> 2, 2 -> 114, 3 -> 4, 4 -> 233.
- A107790 (program): Ones order in the tribonacci substitution of three symbols.
- A107791 (program): Twos order in the tribonacci substitution of three symbols.
- A107792 (program): Threes order in the tribonacci substitution of three symbols.
- A107793 (program): Differences between successive indices of 1’s in the ternary tribonacci sequence A305390.
- A107795 (program): First differences of indices of 2’s in A305389.
- A107796 (program): First differences of indices of 3’s in A305389.
- A107800 (program): a(n) = number of distinct primes dividing A006049(n) (and dividing (A006049(n)+1).
- A107817 (program): Slowest increasing sequence where 2 consecutive integers sum up to a prime.
- A107818 (program): Slowest increasing sequence where (product of 2 consecutive integers)-1 is prime.
- A107819 (program): Slowest increasing sequence where a(n)+n is prime.
- A107820 (program): a(1)=3, a(2)=5; thereafter a(n) = n+5.
- A107836 (program): Slowest increasing sequence where a(n)+(first digit of a(n)) is prime.
- A107839 (program): a(n) = 5*a(n-1) - 2*a(n-2); a(0)=1, a(1)=5.
- A107840 (program): a(n)= 3*a(n-1) -3*a(n-3) +a(n-4), n>6.
- A107841 (program): Series reversion of x(1-3x)/(1-x).
- A107842 (program): A number triangle of lattice walks.
- A107843 (program): Number of iterations of McCarthy 91 Function until it terminates.
- A107844 (program): Highest value obtained in the recursion of McCarthy’s 91 function until it terminates.
- A107849 (program): Expansion of (1 + x)^2 / ((1 - x^2 - 2*x^3)*(1 + x^4)).
- A107852 (program): Expansion of -x*(x^2+1)*(x+1)^2/((2*x^3+x^2-1)*(x^4+1)).
- A107853 (program): G.f. x*(x-1)*(x+1)^3/((2*x^3+x^2-1)*(x^4+1)).
- A107854 (program): G.f. x*(x^2+1)*(x^3-x-1)/((2*x^3+x^2-1)*(x^4+1)).
- A107857 (program): a(n) = floor[(phi + n mod 2)*a(n-1)], a(1)=1.
- A107858 (program): a(n) = b(k), where b(k) = Fibonacci(n-1) and k = floor( n*(1+sqrt(5))/2 ).
- A107862 (program): Triangle, read by rows, where T(n,k) = C(n*(n-1)/2 - k*(k-1)/2 + n-k, n-k).
- A107863 (program): Column 1 of triangle A107862; a(n) = binomial(n*(n+1)/2 + n, n).
- A107867 (program): Triangle, read by rows, where T(n,k) = C(n*(n-1)/2-k*(k-1)/2+n-k+1,n-k).
- A107868 (program): Column 0 of triangle A107867; a(n) = C( n*(n-1)/2 + n + 1, n).
- A107869 (program): Column 1 of triangle A107867; a(n) = binomial( n*(n+1)/2 + n+1, n).
- A107870 (program): Triangle, read by rows, where T(n,k) = C(n*(n-1)/2-k*(k-1)/2+n-k+2, n-k).
- A107871 (program): Column 0 of triangle A107870; a(n) = C( n*(n-1)/2 + n+2, n).
- A107872 (program): Column 1 of triangle A107870; a(n) = C(n*(n+1)/2 + n+2, n).
- A107873 (program): Triangle, read by rows, where T(n,k) = binomial(n*(n-1)/2 - k*(k-1)/2 + n-k+3, n-k).
- A107874 (program): Column 0 of triangle A107873; a(n) = C( n*(n-1)/2 + n+3, n).
- A107875 (program): Column 1 of triangle A107873; a(n) = C( n*(n+1)/2 + n+3, n).
- A107891 (program): a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(3n^2 + 15n + 20)/2880.
- A107895 (program): Euler transform of n!.
- A107903 (program): Generalized NSW numbers.
- A107904 (program): Expansion of (1+6x)/(1-12x^2).
- A107905 (program): Decimal expansion of (5+sqrt(21))/2.
- A107906 (program): Expansion of (1+8x)/(1-16x^2).
- A107907 (program): Numbers having consecutive zeros or consecutive ones in binary representation.
- A107908 (program): a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(3*n+5)/720.
- A107915 (program): a(n) = binomial(n+4,4)*binomial(n+5,4)*binomial(n+6,4)/75.
- A107916 (program): a(n) = binomial(n+3,2)*binomial(n+4,3)*binomial(n+5,5)/12.
- A107917 (program): a(n) = (n+1)(n+2)^2*(n+3)^3*(n+4)^2*(n+5)(n^2 + 6n + 10)/86400.
- A107920 (program): Lucas and Lehmer numbers with parameters (1 +- sqrt(-7))/2.
- A107929 (program): Smallest list of integers from 1 to n such that sum of any two adjacent terms is a square.
- A107941 (program): a(n) = (n+1)(n+2)^2*(n+3)^2*(n+4)^2*(n+5)(3n^2 + 13n + 15)/43200.
- A107942 (program): a(n) = (n+1)(n+2)^3*(n+3)^3*(n+4)(2n+5)/4320.
- A107943 (program): a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)*(2n+3)/8640.
- A107946 (program): Start with S(0)={1}, then S(k+1) equals the concatenation of S(k) with the partial sums of S(k); the limit gives this sequence.
- A107947 (program): Partial sums of A107946.
- A107953 (program): Number of chains in the power set lattice of an (n+3)-element set X_(n+3) of specification n^1 2^1 1, that is, n identical objects of one kind, 2 identical objects of another kind and one other kind. It is the same as the number of fuzzy subsets X_(n+3).
- A107954 (program): Number of chains in the power set lattice, or the number of fuzzy subsets of an (n+4)-element set X_(n+4) with specification n elements of one kind, 3 elements of another and 1 of yet another kind.
- A107955 (program): Number of chains in the power set lattice or the number of fuzzy subsets of an (n+5)-element set X_(n+5) with specification n elements of one kind, 4 elements of another and 1 of yet another kind.
- A107957 (program): Partial sums of A107947.
- A107959 (program): a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)*(n^2 + 5*n + 5)/720.
- A107960 (program): Numbers n such that 11*n - 1 is prime.
- A107961 (program): Pythagorean semiprimes: products of two Pythagorean primes (A002313).
- A107962 (program): a(n) = (n+1)(n+2)^2*(n+3)(n+4)(5n^2 + 18n + 15)/720.
- A107963 (program): a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(5*n^2 + 19*n + 15)/360.
- A107965 (program): a(n) = (n+1)(n+2)^2*(n+3)^2*(n+4)(11n^4 + 110n^3 + 439n^2 + 820n + 600)/86400.
- A107966 (program): a(n) = (n+1)(n+2)^3*(n+3)^2*(n+4)(5n^2 + 23n + 30)/8640.
- A107967 (program): a(n) = (n+1)(n+2)^3*(n+3)^2*(n+4)(n^2 + 4n + 5)/1440.
- A107968 (program): a(n) = (n+1)*(n+2)^3*(n+3)^2*(n+4)*(3n+5)/1440.
- A107970 (program): a(n) = (n+1)*(n+2)^3*(n+3)*(2n+3)*(2n+5)/360.
- A107971 (program): a(n) = (n+1)(n+2)(n+3)(35n^3 + 153n^2 + 232n + 120)/720.
- A107972 (program): Triangle read by rows: T(n,k) = (k+1)(k+2)(n+2)(3n-2k+3)/12 for 0<=k<=n.
- A107973 (program): Numbers of the form a^2 + b for a= 21 to 40 and b= 20 to 1 step -1.
- A107978 (program): Products of two primes of the form 4n+3 (A002145).
- A107979 (program): a(n) = 4*a(n-1) + 2*a(n-2) for n>1, with a(0)=2, a(1)=9.
- A107980 (program): Triangle read by rows: T(n,k) = (n+2)*(k+1)*(k+2)*(2*n-k+2)*(2*n-k+3)/24.
- A107982 (program): Gaussian-Pythagorean semiprimes. Products of a prime of the form 2 or 4n+1 (A002313) and a prime of the form 4n+3 (A002145).
- A107983 (program): Triangle read by rows: T(n,k) = (k+1)(n+2)(n+3)(n-k+2)(n-k+1)/12 for 0<=k<=n.
- A107984 (program): Triangle read by rows: T(n,k) = (k+1)*(n+2)*(2n-k+3)*(n-k+1)/6 for 0 <= k <= n.
- A107985 (program): Triangle read by rows: T(n,k) = (k+1)(n+2)(n-k+1)/2 for 0 <= k <= n.
- A107986 (program): Composite numbers of the form p+2 where p is prime.
- A107987 (program): Odd composite numbers of the form p+2 where p is prime.
- A107991 (program): Complexity (number of maximal spanning trees) in an unoriented simple graph with nodes {1,2,…,n} and edges {i,j} if i + j > n.
- A107992 (program): Numbers n such that 11*n - 3 is prime.
- A107994 (program): Numbers n such that 11*n - 2 is prime.
- A107995 (program): Chebyshev polynomial of the second kind U[n,x] evaluated at x=n+2.
- A108012 (program): a(n)= 8*a(n-1) -16*a(n-2) +4*a(n-4).
- A108014 (program): Expansion of (x^2-2*x)/(x^4-x^2+2*x-1).
- A108019 (program): a(n) = (8^n - 1)*4/7.
- A108020 (program): a(n) is the number whose binary representation is the concatenation of n strings of the four digits “1100”.
- A108021 (program): Numbers n whose binary representation is the first Fibonacci(n) binary digits of the pattern 1010101010101010…
- A108027 (program): Numbers n such that 137*n + 139 is prime.
- A108028 (program): Numbers k such that 139*k + 137 is prime.
- A108029 (program): Numbers n such that 149*k + 151 is prime.
- A108030 (program): Numbers k such that 151*k + 149 is prime.
- A108031 (program): Inverse Moebius transform of Lucas numbers (A000032).
- A108032 (program): Triangle T(n,k), 0<=k<=n, read by rows, defined by : T(0,0) = 1, T((n,k) = 0 if n<k or if k<0, T(n,k) = k*T(n-1, k-1) + (2n-2k-1)*T(n-1, k).
- A108033 (program): n!*(3*n^2-13*n+14)/6.
- A108035 (program): Triangle read by rows: n-th row is n-th nonzero Fibonacci number repeated n times.
- A108036 (program): Triangle read by rows: the triangle in A108035 surrounded by a border of 0’s.
- A108037 (program): Triangle read by rows: n-th row is n-th nonzero Fibonacci number repeated n+1 times.
- A108038 (program): Triangle read by rows: g.f. = (x+y+x*y)/((1-x-x^2)*(1-y-y^2)).
- A108039 (program): Replace each entry 2^i(2j+1) of the triangle in A008280 with i and replace 0’s with *’s; then the entries n in the new triangle do not occur beyond diagonal a(n), measured from the left edge and taking the left edge to be diagonal number 1.
- A108040 (program): Reflection of triangle in A008280 in vertical axis.
- A108044 (program): Triangle read by rows: right half of Pascal’s triangle (A007318) interspersed with 0’s.
- A108045 (program): Triangle read by rows: lower triangular matrix obtained by inverting the lower triangular matrix in A108044.
- A108046 (program): Inverse Moebius transform of Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, …
- A108047 (program): Concatenation of the previous pair of numbers, with the first two terms both 1.
- A108051 (program): a(n+1) = 4*(a(n)+a(n-1)) for n>1, a(1)=1, a(2)=6.
- A108053 (program): Maximum number of diagonals of a regular n-gon that meet at a non-center point.
- A108058 (program): Numbers n such that 179*n + 181 is prime.
- A108059 (program): Numbers n such that 181*n + 179 is prime.
- A108060 (program): Numbers n such that 191*n + 193 is prime.
- A108061 (program): Numbers n such that 193*n + 191 is prime.
- A108073 (program): Triangle in A071943 with rows reversed.
- A108077 (program): Largest prime p such that p-1 divides n.
- A108078 (program): Determinant of a Hankel matrix with factorial elements.
- A108079 (program): a(n) = Sum_{i=0..n} C(2n+i,n+i).
- A108080 (program): Sum_{i=0..n} C(2n+i,n-i).
- A108081 (program): a(n) = Sum_{i=0..n} binomial(2*n-i, n+i).
- A108082 (program): Sum_{i=0..n} binomial(2n+i,2i).
- A108086 (program): Triangle, read by rows, where T(0,0) = 1, T(n,k) = (-1)^(n+k)*T(n-1,k) + T(n-1,k-1); a signed version of Pascal’s triangle.
- A108087 (program): Array, read by antidiagonals, where A(n,k) = exp(-1)*Sum_{i>=0} (i+k)^n/i!.
- A108090 (program): Numbers of the form (11^i)*(13^j).
- A108095 (program): Coefficients of series whose square is the weight enumerator of the [8,4,4] Hamming code (see A002337).
- A108099 (program): a(n) = 8n^2 + 8n + 4.
- A108100 (program): (2*n-1)^2+(2*n+1)^2.
- A108103 (program): Fixed point of the square of the morphism: 1->3, 2->1, 3->121, starting with 1.
- A108104 (program): Sequence A000930 with terms repeated.
- A108105 (program): 2^floor(n/5).
- A108118 (program): Integers not divisible by 3 or 4.
- A108120 (program): Floor[n*1/Sin[1]], or Beatty sequence for 1/sin(1).
- A108122 (program): G.f.: (1-2*x^2)/(1-x-2*x^2-x^3).
- A108124 (program): E.g.f. x/(1+sin(x)).
- A108125 (program): Expansion of e.g.f.: x/(1 - log(1+x)).
- A108126 (program): Maximal number of squares of side 1 in an ellipse of semiaxes n,2n.
- A108131 (program): Array read by antidiagonals: A(k,n) = C(n^k, n).
- A108136 (program): a(1)=1; a(2)=1; a(3)=1; a(n) = 3*a(n-1) + 2*a(n-2) + a(n-3).
- A108137 (program): Primes p such that p + 6^k is composite for all k >= 0.
- A108138 (program): n to the power of the smallest prime divisor of n.
- A108139 (program): n to the power of the largest prime divisor of n.
- A108140 (program): a(n) = 4*a(n-1) -3*a(n-2) -2*a(n-3) +a(n-4), n>8.
- A108141 (program): Least k such that the number (n+1)(n+2)(n+3)…(n+k) >= n^n.
- A108143 (program): a(n)= 5*a(n-1) -a(n-2) -a(n-3).
- A108144 (program): Numbers n such that (n-1)/P(n-1) is a power of two > 1, where P(n) is the largest prime factor of n.
- A108146 (program): a(n)= 4*a(n-1) -a(n-2) -a(n-3).
- A108151 (program): a(n) = n^2 + 3*n + 1 if prime or 0 if composite.
- A108152 (program): a(n)= 3*a(n-1) +2*a(n-2) +a(n-3).
- A108153 (program): a(n) = 3*a(n-1) + 2*a(n-2) + a(n-3).
- A108154 (program): a(n) = n^2 - n + 1 if prime else 0.
- A108161 (program): Partial sums of the positive integers n according to the rule: if n is square then add sqrt(n) else add n.
- A108164 (program): Semiprimes p*q where both p and q are primes of the form 6n+1 (A002476).
- A108165 (program): a(n)=a(n-1) +A108173(n+1) -A108173(n).
- A108171 (program): Tribonacci version of A076662 using beta positive real Pisot root of x^3 - x^2 - x - 1.
- A108172 (program): Semiprimes p*q where p is a prime of the form 6n+1 (A002476) and q is a prime of the form 6n-1 (A007528).
- A108173 (program): Let beta = A058265. Sequence gives a(n) = 1 + ceiling((n-1)*beta^2).
- A108174 (program): Partial sums of the positive integers n according to the rule: if n is square then subtract n, otherwise add n.
- A108177 (program): Integers of the form 2^(4n-1) or 2^(4n), n>0 and their immediate neighbors, together with -1, 0 and 1.
- A108178 (program): a(n) = (n+1)(n+2)^2*(n+3)^2*(n+4)(7n^2 + 23n + 20)/2880.
- A108181 (program): Semiprimes of the form 4n + 1.
- A108187 (program): Numbers n such that 11*n - 5 is prime.
- A108188 (program): a(n) = (n-1)*(a(n-1)+a(n-2)+a(n-3)).
- A108189 (program): a(n) = (n-1)*(a(n-2)+a(n-3))
- A108195 (program): a(n) = n^2 + 5*n - 1.
- A108196 (program): Expansion of (x-1)*(x+1) / (8*x^2 + 1 - 3*x + x^4 - 3*x^3).
- A108197 (program): Number of composite numbers between two successive semiprimes.
- A108198 (program): Triangle read by rows: T(n,k) = binomial(2k+2,k+1)*binomial(n,k)/(k+2) (0 <= k <= n).
- A108201 (program): Numbers of the form (5^i)*(12^j), with i, j >= 0.
- A108204 (program): a(n) = 2*(n-1)*a(n-1) -(n-1)*a(n-2) with a(0)=0, a(1)=1.
- A108205 (program): a(n) = 2*(n-1)*a(n-1)+(n-1)*a(n-2) with a(0)=0, a(1)=1.
- A108206 (program): a(n)= 3*(n-1)*a(n-1) +(n-1)*a(n-2), with a(0)=1, a(1)=1.
- A108207 (program): a(n)= 5*(n-1)*a(n-1) -(n-1)*a(n-2), with a(0)=0, a(1)=1.
- A108208 (program): a(n) = 4*(n-1)*a(n-1) -2*(n-1)*a(n-2), with a(0)=0, a(1)=2.
- A108209 (program): a(n) = 5*(n-1)*a(n-1) -2*(n-1)*a(n-2) with a(0)=0, a(1)=2.
- A108210 (program): Let M[n] be the 2 X 2 matrix {{0, -3}, {(n - 1), 5*(n - 1)}} and let v[1] = {0, 1}’, v[n] = M[n]*v[n - 1]’. Then a[n] is the first entry of v[n].
- A108211 (program): a(n) = 16*n^2 + 1.
- A108213 (program): a(0)=44; if n odd, a(n) = a(n-1)/2, otherwise a(n) = 4*a(n-1).
- A108214 (program): Denominator of the O(x^2) term in the Maclaurin series of the square of the Jacobi polynomial P^{a,b}_n(z) about z=1-x for real positive x.
- A108215 (program): 4-almost primes equal to the product of two successive semiprimes.
- A108216 (program): Number of semiprimes between 10n and 10n + 9.
- A108217 (program): a(0) = 1, a(1) = 1, a(n) = n! + (n-2)! for n >= 2.
- A108218 (program): Numbers of the form (11^i)*(12^j), with i, j >= 0.
- A108221 (program): Primes of the form ceiling(sqrt(prime(n))).
- A108225 (program): a(0) = 0, a(1) = 2; for n >= 2, a(n) = (a(n-1) + a(n-2))*(a(n-1) - a(n-2) + 1)/2.
- A108228 (program): a(n) = (A003961(n) - 1)/2, a permutation of the nonnegative integers.
- A108229 (program): n occurs Lucas number L(n) times (A000204).
- A108230 (program): a(1) = 0, a(n) = order of prime A088387(n).
- A108232 (program): Numbers n such that 11*n - 7 is prime.
- A108233 (program): Numbers n such that 11*n + 5 is prime.
- A108245 (program): If n-th prime is 4m - 1, then a(n) = 4m + 1. If n-th prime is 4m + 1, then a(n) = 4m - 1.
- A108246 (program): Number of labeled 2-regular graphs with no multiple edges, but loops are allowed (i.e., each vertex is endpoint of two (usual) edges or one loop).
- A108248 (program): a(n) = ceiling(n/24) + ceiling((n+1)/24).
- A108257 (program): Numbers n such that concatenating n and the sum of factorials of the digits of n produces a prime.
- A108261 (program): 2nd order recursive series having the property that the product of any two adjacent terms is a triangular number, T(b) = b(b+1)/2 where b equals term a(n) of related series A108262.
- A108262 (program): Second order recursive series having the property that the product of any two adjacent terms equals 4 times a triangular number. That is a(n)*a(n+1)= 4*T(c) = 2c(c+1), where c = the term a(n+1) of related series A108261.
- A108263 (program): Triangle read by rows: T(n,k) is the number of short bushes with n edges and k branchnodes (i.e., nodes of outdegree at least two). A short bush is an ordered tree with no nodes of outdegree 1.
- A108269 (program): Numbers of the form (2*m - 1)*4^k where m >= 1, k >= 1.
- A108281 (program): Numbers that are both triangular and pentagonal of the second kind.
- A108282 (program): a(n) = k*a(n-1) + a(n-2) where k = A003842(a); a(0) = 1.
- A108283 (program): Triangle read by rows, generated from (…, 3, 2, 1).
- A108285 (program): Triangle read by rows, generated from (1, 2, 3, …).
- A108286 (program): Triangle read by rows; columns are simple recursive sequences.
- A108288 (program): Main diagonal of table A060543; a(n) = C((n+1)^2-1, n*(n+1)).
- A108289 (program): Antidiagonal sums of table A060543.
- A108292 (program): Row sums of triangle A108290.
- A108294 (program): a(n) = the least prime p such that p-6n-3 is a power of 2 and 2p-6n-3 is prime, or -1 if no such prime exists.
- A108295 (program): Values of 2p-6n-3 associated with A108294.
- A108296 (program): Diagonal sums of the number triangle associated to A086617.
- A108299 (program): Triangle read by rows, 0 <= k <= n: T(n,k) = binomial(n-[(k+1)/2],[k/2])*(-1)^[(k+1)/2].
- A108300 (program): a(n+2) = 3*a(n+1) + a(n), with a(0) = 1, a(1) = 5.
- A108302 (program): Concatenate n and the sum of the digits of n raised to their own power (A045503).
- A108306 (program): Expansion of (3*x+1)/(1-3*x-3*x^2).
- A108308 (program): Expansion of 1/(1-x^2*c(2x)), c(x) the g.f. of A000108.
- A108309 (program): Consider the triangle of odd numbers where the n-th row has the next n odd numbers. The sequence is the number of primes in the n-th row.
- A108313 (program): Sum of primes q with prime(n) < q < 2*prime(n).
- A108314 (program): Sum of primes p with n^2 < p < (n+1)^2.
- A108319 (program): Numbers of the form (2^i)*(3^j)*(7^k), with i, j, k >= 0.
- A108321 (program): a(n) = n^2 if n^2 is not the difference of two primes; otherwise a(n) = 0.
- A108340 (program): A083952 read mod 2.
- A108341 (program): Numbers n such that 997*n - 1009 is prime.
- A108342 (program): Numbers n such that 1009*n - 997 is prime.
- A108347 (program): Numbers of the form (3^i)*(5^j)*(7^k), with i, j, k >= 0.
- A108350 (program): Number triangle T(n,k) = Sum_{j=0..n-k} binomial(k,j)*binomial(n-j,k)*((j+1) mod 2).
- A108351 (program): Diagonal sums of symmetric triangle A108350.
- A108354 (program): Expansion of 1/((1-x)^2(1+x^2)^2) in powers of x.
- A108355 (program): Expansion of (1+2x^2)/((1-x)^2(1+x^2)^2).
- A108356 (program): Count, repeating multiples of 3 four times, all other numbers twice.
- A108357 (program): Expansion of (1+x^2+x^4)/(1-x^8).
- A108360 (program): Expansion of (1-2x)^2/((1-x)^2(1-2x-x^2)^2).
- A108362 (program): Pair reversal of Fibonacci numbers.
- A108366 (program): L(n,n), where L is defined as in A108299.
- A108367 (program): L(n,-n), where L is defined as in A108299.
- A108368 (program): Coefficients of x/(1-3*x-3*x^2-x^3).
- A108369 (program): Coefficients of x/(1+3*x+3*x^2-x^3).
- A108396 (program): Triangle read by rows: T(n,k) = n*(1+n^k)/2, 0<=k<=n.
- A108397 (program): Sums of rows of the triangle in A108396.
- A108398 (program): a(n) = n*(1 + n^n)/2.
- A108400 (program): a(n) = Product_{k = 0..n} (2^k * k!).
- A108404 (program): Expansion of (1-4x)/(1-8x+11x^2).
- A108411 (program): a(n) = 3^floor(n/2). Powers of 3 repeated.
- A108412 (program): Expansion of (1 + x + x^2)/(1 - 4x^2 + x^4).
- A108413 (program): Expansion of (1+x+5x^2+2x^3) / (1-4x^2+x^4).
- A108414 (program): Number of integer k:s for which max{x^(k-x) | x integer, 0<x<k} = n^(k-n).
- A108415 (program): a(n) = 1, 2 or 3 (resp.) if prime(n) is weak, balanced or strong (resp.).
- A108424 (program): Number of paths from (0,0) to (3n,0) that stay in the first quadrant, consist of steps u=(2,1), U=(1,2), or d=(1,-1) and do not touch the x-axis, except at the endpoints.
- A108426 (program): Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have k peaks of the form Ud.
- A108427 (program): Number of peaks of the form Ud in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1).
- A108429 (program): Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and have k down steps (d).
- A108430 (program): Number of d steps in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1).
- A108432 (program): Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have no hills (a hill is either a ud or a Udd starting at the x-axis).
- A108434 (program): Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have no hills of the form ud (a hill is either a ud or a Udd starting at the x-axis).
- A108436 (program): Number of returns to the x-axis in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1).
- A108440 (program): Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and having k u=(2,1) steps among the steps leading to the first d step.
- A108442 (program): Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) and having only u steps among the steps leading to the first d step.
- A108447 (program): Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and have no peaks of the form ud.
- A108448 (program): Number of peaks of the form ud in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1).
- A108449 (program): Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and having no pyramids (a pyramid is a sequence u^pd^p or U^pd^(2p) for some positive integer p, starting at the x-axis).
- A108450 (program): Number of pyramids in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,-1) (a pyramid is a sequence u^pd^p or U^pd^(2p) for some positive integer p, starting at the x-axis).
- A108452 (program): Number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) and having no pyramids of the first kind (a pyramid of the first kind is a sequence u^pd^p for some positive integer p, starting at the x-axis).
- A108453 (program): Number of pyramids of the first kind in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1),U=(1,2), or d=(1,-1) (a pyramid of the first kind is a sequence u^pd^p for some positive integer p, starting at the x-axis).
- A108474 (program): Expansion of 1/((1-2x)*(1+4x^2)).
- A108475 (program): Expansion of (1-3*x) / (1-5*x-5*x^2+x^3).
- A108476 (program): Expansion of (1-4x)/(1-6x-12x^2+8x^3).
- A108477 (program): A symmetric number triangle based on 2^n.
- A108478 (program): Diagonal sums of number triangle A108477.
- A108479 (program): Diagonal sums of number triangle A086645.
- A108480 (program): Expansion of (1-x-2x^2)/(1-2x-3x^2-4x^3+4x^4).
- A108484 (program): Sum binomial(2n-2k,2k)3^k, k=0..floor(n/2).
- A108485 (program): Sum binomial(2n-2k,2k)2^(n-k), k=0..floor(n/2).
- A108486 (program): Sum binomial(2n-2k,2k)3^k*2^(n-k), k=0..floor(n/2).
- A108487 (program): Sum binomial(2n-2k,2k)10^(n-k), k=0..floor(n/2).
- A108488 (program): Expansion of 1/sqrt(1 -2*x -3*x^2 -4*x^3 +4*x^4).
- A108489 (program): Expansion of 1/sqrt(1-2x-5x^2-6x^3+9x^4).
- A108490 (program): Expansion of 1/sqrt(1-4x-8x^2-24x^3+36x^4).
- A108494 (program): Expansion of f(-q) / f(q) in powers of q where f() is a Ramanujan theta function.
- A108495 (program): a(n) = (n^7 - n)/6.
- A108497 (program): Triangle read by rows: T(n,k) = k^(n+1)-k mod n, showing 1<=k<=n.
- A108498 (program): Triangle read by rows: T(n,k) = sum_i{1<=i<=n} k^i mod n, showing 1<=k<=n.
- A108499 (program): Number of values of k (1<=k<=n) where k^(n+1) = k mod n, or equivalently where sum_i{1<=i<=n} k^i = 0 mod n.
- A108500 (program): Number of values of k (1<=k<=n) where k^(n+1) != k mod n, or equivalently where sum_i{1<=i<=n} k^i != 0 mod n.
- A108513 (program): Numbers of the form (2^i)*(5^j)*(7^k), with i, j, k >= 0.
- A108514 (program): If n is a power of 2, a(n)=n; otherwise a(n) = (p-1)*n/p where p = smallest odd prime divisor of n.
- A108520 (program): Expansion of 1/(1+2*x+2*x^2).
- A108524 (program): Number of ordered rooted trees with n generators.
- A108546 (program): Lexicographically earliest permutation of primes such that for n>1 forms 4*k+1 and 4*k+3 alternate.
- A108552 (program): Integer values of (1*2*…*k)/(1+2+…+k) = k!/T(k) = A000142(k)/A000217(k), k>=1.
- A108561 (program): Triangle read by rows: T(n,0)=1, T(n,n)=(-1)^n, T(n+1,k)=T(n,k-1)+T(n,k) for 0 < k < n.
- A108562 (program): Primes of the form prime(n) + prime(n+1) - 2n - 1.
- A108568 (program): a(n) = prime(n) + prime(n+1) - 2n - 1.
- A108570 (program): Squares of lesser of twin primes.
- A108576 (program): Number of 3 X 3 magic squares (with distinct positive entries) having all entries < n.
- A108577 (program): Number of symmetry classes of 3 X 3 magic squares (with distinct positive entries) having all entries < n.
- A108578 (program): Number of 3 X 3 magic squares with magic sum 3n.
- A108579 (program): Number of symmetry classes of 3 X 3 magic squares (with distinct positive entries) having magic sum 3n.
- A108580 (program): Numbers whose sum of bits when written in binary > sum of decimal digits.
- A108581 (program): Positive triangular numbers repeated their own number of times.
- A108582 (program): n appears n^3 times.
- A108584 (program): Numbers k such that 10*k - 97 is prime.
- A108586 (program): Floor(2*n*Pi/(2*Pi-1)).
- A108587 (program): Floor(n/(1-sin(1))).
- A108588 (program): Numbers n such that 10*n + 97 is prime.
- A108589 (program): Floor(n*Pi/(Pi-2)).
- A108594 (program): Numbers n such that 10*n + 101 is prime.
- A108595 (program): Numbers n such that 10*n + 103 is prime.
- A108598 (program): Floor(n*((5+sqrt(5))/4)).
- A108600 (program): Number of freely braided permutations of length n; the freely braided permutations are those that avoid 3421, 4231, 4312 and 4321.
- A108601 (program): Numbers n such that 7*n - 911 is prime.
- A108604 (program): Squares of the form prime(k)*prime(k+1) + 2*prime(k+1).
- A108605 (program): Semiprimes with prime sum of factors: twice the lesser of the twin prime pairs.
- A108611 (program): Excess of Beatty-function of 1/sin(1) over n.
- A108612 (program): Beatty-2 (or nested Beatty) sequence for 1/sin(1).
- A108613 (program): Excess of Beatty-2 function of 1/sin(1) over n^2.
- A108623 (program): G.f. satisfies x = (A(x)-(A(x))^2)/(1-A(x)-(A(x))^2).
- A108624 (program): G.f. satisfies x = (A(x)+(A(x))^2)/(1-A(x)-(A(x))^2).
- A108625 (program): Square array, read by antidiagonals, where row n equals the crystal ball sequence for A_n lattice.
- A108626 (program): Antidiagonal sums of square array A108625, in which row n equals the crystal ball sequence for A_n lattice.
- A108627 (program): Logarithmic g.f.: Sum_{n>=1} a(n)/n*x^n = log(G108626(x)), where G108626(x) is g.f. for A108626.
- A108628 (program): n-th term of the crystal ball sequence for A_{n+1} lattice for n >= 0.
- A108632 (program): Semiprimes with prime digits (only digits 2,3,5,7 in semiprimes).
- A108634 (program): Semiprimes with ordered digits.
- A108636 (program): Semiprimes with even digits.
- A108644 (program): Square table T(n,n) read by ascending antidiagonals: T(i,i)=i*i, if i>j T(i,j)=i*(i-1)+j, if j>i T(i,j)=(j-1)*(j-1)+i.
- A108645 (program): a(n) = (n+1)(n+2)^2*(n+3)^2*(n+4)(2n^2 + 6n + 5)/720.
- A108647 (program): a(n) = (n+1)^2*(n+2)^2*(n+3)^2*(n+4)/144.
- A108648 (program): a(n) = (n+1)^2*(n+2)^3*(n+3)/24.
- A108649 (program): a(n) = (n+1)(n+2)(n+3)(13n^3 + 69n^2 + 113n + 60)/360.
- A108650 (program): a(n) = (n+1)^2*(n+2)*(n+3)*(3*n+4)/24.
- A108651 (program): Multiples of 8 that are divisible by no prime > 5.
- A108662 (program): Numbers whose sum of squares of digits is a prime.
- A108666 (program): Number of (1, 1)-steps in all Delannoy paths of length n.
- A108667 (program): Triangle read by rows: T(n,k) = 9k*n + 14(n+k) + 20 (0 <= k <= n).
- A108669 (program): Triangle read by rows: T(n,k) = 11*k*n + 14*(n+k) + 20 (0 <= k <= n).
- A108670 (program): a(n) = (n+1)(n+2)^3*(n+3)(n+4)(5n^2 + 16n + 15)/1440.
- A108671 (program): a(n) = (n+1)(n+2)^3*(n+3)(13n^2 + 37n + 30)/720.
- A108673 (program): a(n) = (n+1)(n+2)(n+3)(2n+3)(10n^2 + 27n + 20)/360.
- A108674 (program): a(n) = (n+1)^2 * (n+2)^2 * (2*n+3) / 12.
- A108675 (program): a(n) = (n+1)*(n+2)*(61*n^4 + 366*n^3 + 845*n^2 + 888*n + 360)/720.
- A108676 (program): a(n) = (n+1)^2*(n+2)*(5*n^2 + 15*n + 12)/24.
- A108677 (program): a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(19*n^2 + 47*n + 30)/720.
- A108678 (program): a(n) = (n+1)^2*(n+2)*(2n+3)/6.
- A108679 (program): a(n) = (n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)^2*(n+6)/86400.
- A108680 (program): Kekulé numbers for certain benzenoids.
- A108681 (program): a(n) = (n+1)*(n+2)^2*(n+3)*(n+4)*(n+5)*(2*n+3)/720.
- A108682 (program): a(n) = (n+1)*(n+2)^2*(n+3)*(n+4)*(4*n^2+15*n+15)/720.
- A108683 (program): a(n) = (n+1)*(n+2)*(n+3)*(n+4)*(7*n^2 + 20*n + 15)/360.
- A108684 (program): a(n) = (n+1)*(n+2)*(n+3)*(19*n^3 + 111*n^2 + 200*n + 120)/720.
- A108689 (program): Smallest integer q >= 1 such that difference between q*Pi and the nearest integer is <= 1/n.
- A108696 (program): Generated by a sieve: see comments.
- A108704 (program): Number of partitions of 112233…nn into n pairs.
- A108711 (program): Number of partitions of n with floor(2n/3) parts.
- A108713 (program): Number of possible canonical minimal transition-complete sequences over n objects.
- A108715 (program): First differences of A025480.
- A108719 (program): Primes which can be partitioned into a sum of distinct primes in more than one way.
- A108720 (program): a(n)=a(n-1)+sum of digits(a(n-1))*sum of digits(a(n-2)).
- A108724 (program): Numbers n such that 11*n + 17 is prime.
- A108725 (program): Numbers n such that 11*n + 19 is prime.
- A108726 (program): Numbers n such that 11*n + 29 is prime.
- A108727 (program): Numbers n such that 11*n + 31 is prime.
- A108732 (program): a(0)=22; if n odd, a(n) = a(n-1)/2, otherwise a(n) = 4*a(n-1).
- A108733 (program): Expansion of (1+18*x)^(1/3).
- A108734 (program): Expansion of (1 + 24*x)^(1/2).
- A108735 (program): Expansion of sqrt(1 + 12*x).
- A108738 (program): a(n) = n/(smallest odd prime divisor of n), if any.
- A108741 (program): Member r=100 of the family of Chebyshev sequences S_r(n) defined in A092184.
- A108742 (program): Row sums of a triangle related to the Jacobsthal polynomials.
- A108744 (program): Decimal expansion of B = Sum_{ n > 0 } 1/A007559(n).
- A108747 (program): Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n and having k returns to the x-axis.
- A108751 (program): Numbers n such that 11*n - 911 is prime.
- A108752 (program): Numbers k such that 12 divides k*(k+1).
- A108753 (program): Difference between the n-th partial sum of the squares (A000330) and the n-th partial sum of the primes (A007504).
- A108754 (program): Difference between partial sum of the first n primes and n^2.
- A108756 (program): A triangle related to the Jacobsthal polynomials.
- A108757 (program): Numbers n such that 1000n + 911 is prime.
- A108758 (program): a(n) = 2*a(n-1) - a(n-4) + a(n-5) with a(-1)=a(0)=a(1)=1, a(2)=2, a(3)=4, a(4)=7.
- A108762 (program): Numbers n such that 911*n + 13 is prime.
- A108763 (program): If n-th prime is 8m+1, then a(n) = 8m+3. If n-th prime is 8m+3, then a(n) = 8m+5. If n-th prime is 8m+5, then a(n) = 8m+7. If n-th prime is 8m+7, then a(n) = 8m+1.
- A108765 (program): G.f. (1 - x + x^2)/((1-3*x)*(x-1)^2).
- A108766 (program): a(n) = A007290(n+2) - 1 = 2*C(n+2,3) - 1.
- A108767 (program): Triangle read by rows: T(n,k) is number of paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(1,1), d=(1,-2) and have k peaks (i.e., ud’s).
- A108769 (program): Numbers m such that m^2 + (m+1)^2 is a semiprime.
- A108771 (program): Numbers of the form (12^i)*(13^j), with i, j >= 0.
- A108773 (program): Concatenation of n and the sum of the digits of n.
- A108775 (program): a(n) = floor(sigma(n)/n).
- A108782 (program): Difference between n and the largest number with the same digit set as n.
- A108784 (program): Difference between A107757 and A107755.
- A108786 (program): Yet another version of the Catalan triangle A008315.
- A108791 (program): a(2n) = -5*(fibonacci(6n+2))^2, a(2n+1) = (lucas(6n+5))^2.
- A108793 (program): Semiprimes that can be partitioned into a sum of semiprimes in more than one way.
- A108803 (program): A108802 read mod 4.
- A108804 (program): Self-convolution of A010060 (Thue-Morse sequence).
- A108805 (program): A108804 read mod 4.
- A108812 (program): 11^n mod 50.
- A108813 (program): Decimal expansion of the continued fraction 2/(5+4/(7+6/(9+8/(11+10/(13+12/…
- A108815 (program): Indices of triangular numbers which are products of 3 primes.
- A108838 (program): Triangle of Dyck paths counted by number of long interior inclines.
- A108850 (program): Number of 1’s in the binary expansion of the repunits.
- A108851 (program): a(n) = 4*a(n-1) + 3*a(n-2), a(0) = 1, a(1) = 2.
- A108852 (program): Number of Fibonacci numbers <= n.
- A108854 (program): Numbers n such that 10*n - 127 is prime.
- A108855 (program): Numbers n such that 10*n + 127 is prime.
- A108856 (program): Numbers n such that 10*n - 131 is prime.
- A108857 (program): Numbers n such that 10*n + 131 is prime.
- A108863 (program): Number of Dyck paths containing exactly one UUUD.
- A108865 (program): Numbers n such that the perfect deficiency of n (A109883) is prime.
- A108866 (program): Numerator of Sum_{k=1..n} 2^k/k.
- A108869 (program): E.g.f. : exp(6x)/(1-x).
- A108870 (program): Tokuda’s good set of increments for Shell sort.
- A108872 (program): Sums of ordinal references for a triangular table read by columns, top to bottom.
- A108873 (program): Numbers n whose base 3 representations, interpreted as base 10 integers, are semiprimes.
- A108874 (program): Numbers n such that 41*n + 43 is prime.
- A108882 (program): Period doubling sequence starting with ‘1 0 1’.
- A108891 (program): Triangle read by rows: T(n,k) = number of Schroeder (or royal) n-paths (A006318) containing k returns to the diagonal y=x. (A northeast step lying on y=x contributes a return.)
- A108895 (program): Partial sums of quadruple factorial numbers n!!!! (A007662).
- A108896 (program): Numbers whose outer two digits are 9’s and inner digits are 4’s.
- A108898 (program): a(n+3) = 3*a(n+2) - 2*a(n), a(0) = -1, a(1) = 1, a(2) = 3.
- A108899 (program): Numbers n such that 11*n + 2357 is prime.
- A108902 (program): Numbers n such that 23*n + 2357 is prime.
- A108903 (program): Numbers such that the outer 2 digits are 9 and the inner digits are 5.
- A108904 (program): Numbers such that the outer two digits are 9’s and the inner digits are 7’s.
- A108906 (program): First differences of A006899.
- A108908 (program): Largest n-digit number coprime to n.
- A108909 (program): Sum of k-digit multiples of n where k is the number of digits in n.
- A108911 (program): Difference between n and the sum of the factorials of its digits.
- A108916 (program): Triangle of Schroeder paths counted by number of diagonal steps not preceded by an east step.
- A108918 (program): Reversed binary words in reversed lexicographic order.
- A108919 (program): Number of series-reduced labeled trees with n nodes.
- A108920 (program): Number of positive integers k>n such that n+k divides n^2+k^2.
- A108922 (program): Expansion of 1/((x^8+1)*(x-1)^2).
- A108923 (program): Expansion of 1/((x^8+1)*(1-x)^3).
- A108924 (program): J(n)^2+J(n+1)^2, with J(n) the Jacobsthal number A001045(n).
- A108928 (program): a(n) = 8*n^2 - 3.
- A108929 (program): Expansion of (1-x^4-2*x^3)/((x-1)*(x^2+x+1)*(x^2+4*x-1)).
- A108931 (program): a(2n) = -A106328(n), a(2n+1) = A054488(n).
- A108935 (program): Numbers n such that 7*n + 911 is prime.
- A108936 (program): Numbers n such that 11*n + 911 is prime.
- A108937 (program): Numbers n such that 911*n + 11 is prime.
- A108938 (program): Numbers n such that 911*n + 7 is prime.
- A108942 (program): Degrees of irreducible representations of SL(2,7).
- A108943 (program): Square root of A108945(n).
- A108945 (program): Squares equal to the sum of the n-th twin prime pair and minimal square k^2.
- A108946 (program): a(2n) = A001570(n), a(2n+1) = -A007654(n+1).
- A108951 (program): Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).
- A108953 (program): Convolution of 3^n*n! and n!.
- A108954 (program): a(n) = pi(2*n) - pi(n). Number of primes in the interval (n,2n].
- A108955 (program): Floor(Li(2n) - Li(n)).
- A108956 (program): Floor(R(2n) - R(n)).
- A108958 (program): Number of unordered pairs of distinct length-n binary words having the same number of 1’s.
- A108964 (program): Write n in balanced ternary notation, omit any zeros and form the left-to-right alternating sum mod 3.
- A108969 (program): Numbers n such that 43*n + 41 is prime.
- A108975 (program): Product of all primes with primitive root 2 less than or equal to some prime with primitive root 2.
- A108976 (program): Numbers n such that 17*n + 19 is prime.
- A108977 (program): Numbers n such that 19*n + 17 is prime.
- A108978 (program): Numbers k such that 29*k + 31 is prime.
- A108979 (program): Numbers n such that 31*n + 29 is prime.
- A108980 (program): Numbers k such that the string k9111 is prime.
- A108981 (program): a(n) = 3a(n-1) + 4a(n-2), a(0) = 1, a(1) = 5.
- A108982 (program): Inverse binomial of A003949.
- A108983 (program): Inverse binomial transform of A003950.
- A108984 (program): Inverse binomial transform of A003951.
- A108985 (program): Expansion of (x+1)*(x^3-x^2-x-1)/((1-x)*(x^2+2*x-1)*(x^2+x+1)).
- A109001 (program): Triangle, read by rows, where g.f. of row n equals the product of (1-x)^n and the g.f. of the coordination sequence for root lattice B_n, for n >= 0.
- A109002 (program): Maximal difference between two n-digit numbers.
- A109004 (program): Table of gcd(n,m) read by antidiagonals, n >= 0, m >= 0.
- A109007 (program): a(n) = gcd(n,3).
- A109008 (program): a(n) = gcd(n,4).
- A109009 (program): a(n) = gcd(n,5).
- A109010 (program): a(n) = gcd(n,7).
- A109011 (program): a(n) = gcd(n,8).
- A109012 (program): a(n) = gcd(n,9).
- A109013 (program): a(n) = gcd(n,10).
- A109014 (program): a(n) = gcd(n,11).
- A109015 (program): a(n) = gcd(n,12).
- A109016 (program): Concatenate n and the sum of factorials of the digits of n.
- A109017 (program): a(n) = Kronecker symbol (-6/n).
- A109020 (program): (2*7^n - 3*3^n + 1)/6.
- A109021 (program): (2*7^n - 6*3^n + 4)/6.
- A109033 (program): Number of permutations in S_n avoiding the patterns 1342 and 2143.
- A109034 (program): First differences of A109033.
- A109039 (program): Expansion of eta(q) * eta(q^3) * (eta(q^4) * eta(q^6) / eta(q^12))^2 in powers of q.
- A109040 (program): Expansion of 1-eta(q)eta(q^3)(eta(q^4)eta(q^6))^2/eta(q^12)^2 in powers of q.
- A109041 (program): Expansion of eta(q)^9 / eta(q^3)^3 in powers of q.
- A109042 (program): Table read by antidiagonals: T(n,m) = lcm(n,m) (n >= 0, m >= 0).
- A109043 (program): a(n) = lcm(n,2).
- A109044 (program): a(n) = lcm(n,3).
- A109045 (program): a(n) = lcm(n,4).
- A109046 (program): a(n) = lcm(n, 5).
- A109047 (program): a(n) = lcm(n, 6).
- A109048 (program): a(n) = lcm(n, 7).
- A109049 (program): a(n) = lcm(n, 8).
- A109050 (program): a(n) = lcm(n, 9).
- A109051 (program): a(n) = lcm(n,10).
- A109052 (program): a(n) = lcm(n,11).
- A109053 (program): a(n) = lcm(n,12).
- A109064 (program): Expansion of eta(q)^5 / eta(q^5) in powers of q.
- A109065 (program): Numerator of the fraction due in month n of the total interest for a one-year installment loan based on the Rule of 78s (each denominator is 78).
- A109066 (program): Number of prime digits in n-th prime.
- A109067 (program): 3-almost primes of the form semiprime + 1.
- A109075 (program): Number of primes which use each of 0-to-n decimal digits exactly once.
- A109078 (program): Number of symmetric Dyck paths of semilength n and having no hills (i.e., no peaks at level 1).
- A109081 (program): Reversion of x*(1-x)*(1-x^2)*(1-x^3)/(1-x^6).
- A109082 (program): Depth of rooted tree having Matula-Goebel number n.
- A109083 (program): Convolution of A002324 and A010815.
- A109091 (program): Expansion of (1 - eta(q)^5 / eta(q^5)) / 5 in powers of q.
- A109094 (program): Length of the longest path (repeated edges not allowed) between two arbitrary, distinct nodes in K_n, the complete graph on n vertices.
- A109105 (program): a(n) = (8*sqrt(5)/25)((sqrt(5) + 2)((15 + 5*sqrt(5))/2)^n + (sqrt(5) - 2)((15 - 5*sqrt(5))/2)^n.
- A109106 (program): a(n) = (1/sqrt(5))*((sqrt(5) + 1)*((15 + 5*sqrt(5))/2)^(n-1) + (sqrt(5) - 1)*((15 - 5*sqrt(5))/2)^(n-1)).
- A109107 (program): a(n) = (1/sqrt(26))((5+sqrt(26))^(n+1)-(5-sqrt(26))^(n+1)).
- A109108 (program): a(n) = 10a(n-1) + a(n-2), a(0)=1, a(1)=9.
- A109109 (program): a(0)=1, a(1)=4, a(n) = 10a(n-1) + a(n-2).
- A109110 (program): a(n) = 2a(n-1) + a(n-2) - a(n-3); a(0)=4, a(1)=9, a(2)=20.
- A109112 (program): a(n) = 6*a(n-1) - 3*a(n-2), a(0)=2, a(1)=13.
- A109113 (program): a(n) = 6a(n-1) + 3a(n-2), a(0)=2, a(1)=14.
- A109114 (program): a(n) = 5*a(n-1) - 3*a(n-2), a(0)=1, a(1)=6.
- A109115 (program): a(n) = 4*a(n-1) + 3*a(n-2), a(0)=1, a(1)=6.
- A109116 (program): a(n) = (n+1)^3*(n+2)^2*(n+5).
- A109117 (program): a(n) = (n+1)^3*(2n+1)(5n+1).
- A109118 (program): a(n) = 2*(n^2 + 3*n + 1)^3.
- A109119 (program): a(n) = 2(5n^2 + 5n + 1)^3.
- A109120 (program): a(n) = 10*(n+1)^3*(n+2)*(5*n+7)^2.
- A109121 (program): a(n) = 10(n+1)^3*(2n+1)(7n+5)^2.
- A109123 (program): a(n) = 4*(n+1)^2*(n+3)^2*(5*n^2 + 20*n + 12).
- A109124 (program): a(n) = (n+1)*(n+2)^3*(n+3)^4*(n+4)^3*(n+5)*(2n+5)*(2n+7)/7257600.
- A109127 (program): Expansion of q^(-1/8) * (eta(q^3) - eta(q)^3) / 3 in powers of q.
- A109128 (program): Triangle read by rows: T(n,k) = T(n-1,k-1) + T(n-1,k) + 1 for 0<k<n, T(n,0) = T(n,n) = 1.
- A109129 (program): Width (i.e., number of non-root vertices having degree 1) of the rooted tree with Matula-Goebel number n.
- A109130 (program): Magic constant of smallest order-n perfect magic cube.
- A109133 (program): Numbers k such that (sum of digits)*(number of digits) + 1 is prime.
- A109134 (program): Decimal expansion of Phi, the real root of the equation 1/x = (x-1)^2.
- A109139 (program): Numerators associated with the continued fraction of the differences of consecutive prime numbers.
- A109140 (program): Denominators associated with A109139.
- A109161 (program): Triangle read by rows: T(n, k) = n*(n+9) + k + 5, with T(0, 0) = 5 and T(1, 0) = 15.
- A109163 (program): a(n) = A019565(n-th prime).
- A109164 (program): a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), n >= 3; a(0)=1, a(1)=6, a(2)=20.
- A109165 (program): a(n) = 5*a(n-2) - 2*a(n-4), n >= 4; a(n) = (1/6)*(-1)^n + 4/3)*2^n - 1/2.
- A109168 (program): Continued fraction expansion of the constant x (A109169) such that the continued fraction of 2*x yields the continued fraction of x interleaved with the positive even numbers.
- A109170 (program): Continued fraction expansion of 2*x which equals the continued fraction of x (A109168) interleaved with positive even numbers.
- A109173 (program): Recursive form of A109845 but with a(1)=1.
- A109174 (program): Number of steps to reach 1 in the modified `3x+1’-type problem defined by: If a_n is even then a_(n+1) = a_n/2. If a_n is = 1 (mod 4) then a_(n+1) = 3a_n+1. If a_n is = 3 (mod 4) then a_(n+1) = 3a_n-1.
- A109175 (program): Minimum number of moves to solve the first Panex puzzle of order n of transferring a side tower to the center column.
- A109180 (program): The set N of numbers such that each positive integer can be written in the form F + n, where F is a Fibonacci number and n in N, in an even number of ways.
- A109187 (program): Triangle read by rows: T(n,k) is number of Grand Motzkin paths of length n having k (1,0)-steps.
- A109188 (program): Number of (1,0) steps in all Grand Motzkin paths of length n.
- A109190 (program): Number of (1,0)-steps at level zero in all Grand Motzkin paths of length n.
- A109194 (program): Number of returns to the x-axis (i.e., d or u steps hitting the x-axis) in all Grand Motzkin paths of length n. (A Grand Motzkin path of length n is a path in the half-plane x>=0, starting at (0,0), ending at (n,0) and consisting of steps u=(1,1), d=(1,-1) and h=(1,0).).
- A109196 (program): Number of returns to the x-axis from above (i.e., d steps hitting the x-axis) in all Grand Motzkin paths of length n.
- A109214 (program): Product of a(n-1) and digit reversal of a(n-2).
- A109217 (program): a(n) is the binary string of length n+1 that has 0’s at indices that are squares and 1’s elsewhere, where the most significant digit has index 0.
- A109220 (program): Expansion of (1+x-x^2)/(1-2x-2x^2+x^4).
- A109221 (program): A number triangle related to the Fibonacci polynomials.
- A109222 (program): Row sums of a triangle related to the Fibonacci polynomials.
- A109223 (program): Number triangle related to the Fibonacci polynomials.
- A109225 (program): Triangle read by rows: T(n,0) = T(n,n) = 1 and for 0 < k < n: T(n,k) = T(n-1,k-1) + 1 - T(n-1,k-1) mod 2 + T(n-1,k).
- A109227 (program): Binary strings that have 1’s where primes occur, 0’s elsewhere and every term ends with the n-th prime index.
- A109231 (program): a(n) = floor(n*cosh(1)).
- A109232 (program): a(n) = floor(n*(e^2+1)/(e-1)^2).
- A109234 (program): a(n) = floor(n*sinh(1)).
- A109235 (program): a(n) = floor(n*(e^2-1)/(e^2-2*e-1)).
- A109237 (program): a(n) = floor(n*coth(1)).
- A109238 (program): a(n) = floor(n*(e^2+1)/2).
- A109241 (program): Expansion of 1/((1-10*x)*(1-100*x)).
- A109242 (program): Expansion of 1/((1-x)(1-10x)(1-100x)).
- A109244 (program): A tree-node counting triangle.
- A109246 (program): Riordan array (1-x-2x^2,x(1-x)).
- A109247 (program): Expansion of (1 - 3*x^2 - 3*x^3 + x^4)/(1 + x^4).
- A109248 (program): Expansion of (1-x-2*x^2)/(1-x^2+x^3).
- A109253 (program): Number of elements of the Weyl group of type B where a reduced word contains all of the simple reflections.
- A109255 (program): a(n) = (p^2 - 1) / 12, where p is the n-th prime of the form 4*k+1.
- A109256 (program): a(n) = n^6 - 11n^4 + 36n^2 - 36.
- A109259 (program): a(n) = floor(n*sqrt(2)^sqrt(2)).
- A109260 (program): a(n) = floor(n*sqrt(2)^sqrt(2)/(sqrt(2)^sqrt(2)-1)).
- A109262 (program): A Catalan transform of the Fibonacci numbers.
- A109263 (program): A Catalan transform of F(n-1)-0^n.
- A109264 (program): Riordan array (1-x-x^2,x(1-x)).
- A109265 (program): Row sums of Riordan array (1-x-x^2,x(1-x)).
- A109266 (program): Diagonal sums of Riordan array (1-x-x^2,x(1-x)).
- A109267 (program): Riordan array (1/(1 - x*c(x) - x^2*c(x)^2), x*c(x)) where c(x) is the g.f. of A000108.
- A109269 (program): Numbers n such that n^2 < (1/2)*(prevprime(n^2)+nextprime(n^2).
- A109270 (program): Numbers n such that n^2 > (1/2)(prevprime(n^2)+nextprime(n^2)).
- A109274 (program): Numbers n such that n+1 is prime, 2n+1 composite.
- A109288 (program): Semiprimes equal to p*q + 1, where p and q are distinct primes.
- A109301 (program): a(n) = rhig(n) = rote height in gammas of n, where the “rote” corresponding to a positive integer n is a graph derived from the primes factorization of n, as illustrated in the comments.
- A109313 (program): Difference between prime factors of n-th semiprime.
- A109325 (program): Zsigmondy numbers for a = 3, b = 2: Zs(n, 3, 2) is the greatest divisor of 3^n - 2^n (A001047) that is relatively prime to 3^m - 2^m for all positive integers m < n.
- A109338 (program): Triangle read by rows: T(n,k) = number of inequivalent binary sequences of length n and weight k, where two sequences are said to be equivalent if they have the same set of phrases in their Ziv-Lempel encodings (the phrases can appear in a different order in the two sequences).
- A109340 (program): Expansion of x^2*(1+x+4*x^2)/((1+x+x^2)*(1-x)^3).
- A109341 (program): Take a deck of 52 cards face-down, split it in half and flip one deck and reinsert it into the other deck such that the cards are alternatingly face up and face down. This sequence is the number of face-up cards after repeating this process n times.
- A109344 (program): a(n) consists of n 4’s, n-1 8’s and a single 9 (in that order).
- A109345 (program): a(n) = 5^((n^2 - n)/2).
- A109352 (program): a(n) = sum of the prime divisors of the n-th squarefree composite number.
- A109353 (program): a(n) is the sum of the distinct prime divisors of A024619(n).
- A109354 (program): a(n) = 6^((n^2 - n)/2).
- A109358 (program): Square root of squares of form 2*p + 3, where p is prime.
- A109359 (program): Expansion of x*(1+x^2+3*x^3+2*x^4+x^5+4*x^6) / ((x^2+1)*(x^2-x+1)*(x-1)^2*(x+1)^2).
- A109360 (program): Expansion of x*(1+4*x+5*x^2-x^3+6*x^4+x^5-4*x^6) / ((x^2+1)*(x^2-x+1)*(x-1)^2*(x+1)^2).
- A109362 (program): Period 6: repeat [0, 0, 1, 2, 0, 3].
- A109363 (program): a(n) = 4*2^n - 3*n - 5.
- A109365 (program): a(-1)=a(0)=1 and recursively a(n) = prime(n)*(a(n-1)+a(n-2)).
- A109366 (program): a(-1)=0, a(0)=1 and recursively a(n) = prime(n)*(a(n-1)+a(n-2)).
- A109367 (program): Squares of the form 2*p + 3, where p is a prime.
- A109369 (program): Numbers n such that the string 33n is the decimal expansion of a prime number.
- A109370 (program): Numbers n such that the string 22n is the decimal expansion of a prime number.
- A109371 (program): Numbers k such that the string 11k is prime.
- A109372 (program): Numbers k such that k * (sum of the digits of k raised to their own power) + 1 is prime.
- A109373 (program): Semiprimes of the form semiprime + 1.
- A109375 (program): Bisection of A093411.
- A109377 (program): Expansion of ( 2+x+2*x^2 ) / ( 1-2*x+x^2-x^3 ).
- A109381 (program): Maximum digit of n^2 written in factorial base.
- A109386 (program): G.f. is the logarithm of the g.f. of A107742: Sum_{n>=1} (a(n)/n)*x^n = Log( Sum_{n>=0} A107742(n)*x^n ).
- A109388 (program): Maximum number of pairwise incomparable subcubes of the discrete n-cube. Largest antichain in partial ordering {0,1,*)^n where 0 and 1 are less than *. Maximum number of implicants in an irredundant disjunctive normal form for n Boolean variables.
- A109389 (program): Expansion of q^(-1/12)eta(q)eta(q^6)/(eta(q^2)eta(q^3)) in powers of q.
- A109391 (program): a(n) = (n^(n+1))*(n + 1)/2 = A000217(n)*A000312(n).
- A109392 (program): Partial sums of A109391.
- A109394 (program): A000934(A000934(n)).
- A109395 (program): Denominator of phi(n)/n = Product_{p|n} (1 - 1/p); phi(n)=A000010(n), the Euler totient function.
- A109398 (program): a(n) = (1/n!)*Sum_{k=0..n} (n+k)!.
- A109403 (program): Examine the sequence of all (even or odd) semiprimes, A001358, and record the averages of any pair of successive terms of the same parity.
- A109408 (program): Cubes whose digits sum to a prime.
- A109410 (program): Prime numbers p such that p = digit sum of cubes in A109408.
- A109411 (program): Partition the sequence of positive integers into minimal groups so that sum of terms in each group is a semiprime; sequence gives sizes of the groups.
- A109413 (program): a(n) equals the (n*(n+1)/2)-th partial sum of the self-convolution of A010054, which has the g.f.: Sum_{n>=0} x^(n*(n+1)/2).
- A109414 (program): a(n) equals the (n*(n+1)/2)-th partial sum of the self-convolution cube of A010054, which has the g.f.: Sum_{k>=0} x^(k*(k+1)/2).
- A109415 (program): a(n) equals the (n*(n+1)/2)-th partial sum of the self-convolution 4th power of A010054, which has the g.f.: Sum_{k>=0} x^(k*(k+1)/2).
- A109421 (program): Numbers n such that tau(n)/bigomega(n) is an integer [tau(n)=number of divisors of n; bigomega(n)=number of prime divisors of n, counted with multiplicities].
- A109422 (program): Numbers n such that tau(n)/bigomega(n) is not an integer [tau(n) =number of divisors of n; bigomega(n)=number of prime divisors of n, counted with multiplicities].
- A109423 (program): Numbers n such that sigma(n)/bigomega(n) is an integer [sigma(n) = sum of divisors of n; bigomega(n) = number of prime divisors of n, counted with multiplicity].
- A109430 (program): Period 24.
- A109431 (program): Binary strings that have 1’s where ‘evil numbers’ occur, 0’s elsewhere and every term ends with the n-th evil number index (counting with 0 = first).
- A109437 (program): a(-1) = a(0) = 0, a(1) = 1; a(n) = 5a(n-1) - 5a(n-2) + a(n-3) + 2*(-1)^(n+1), alternatively a(n) = 3a(n-1) + 3a(n-2) - a(n-3).
- A109438 (program): a(n) = 5a(n-1) - 5a(n-2) + a(n-3) + 2*(-1)^(n+1), alternatively a(n) = 3a(n-1) + 3a(n-2) - a(n-3).
- A109442 (program): Cumulative sum of smallest prime power >= n.
- A109443 (program): Cumulative sum of largest prime power <= n.
- A109444 (program): Cumulative sum of mosaic numbers (A000026).
- A109446 (program): Binomial coefficients C(n,k) with n-k even, read by rows.
- A109447 (program): Binomial coefficients C(n,k) with n-k odd, read by rows.
- A109449 (program): Triangle read by rows, T(n,k) = binomial(n,k)*A000111(n-k), 0 <= k <= n.
- A109450 (program): Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, …] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …] where DELTA is the operator defined in A084938.
- A109451 (program): a(1)=1; a(n) = smallest positive integer not already present such that a(n-1) and a(n) have a different number of 1’s in their binary expansions.
- A109453 (program): Cumulative sum of initial digits of n.
- A109454 (program): Sum of non-Fibonacci numbers between successive Fibonacci numbers: a(n) = Sum_{k=F(n)+1..F(n+1)-1)} k.
- A109466 (program): Riordan array (1, x(1-x)).
- A109469 (program): Cumulative sum of coefficients of ménage hit polynomials (A000033).
- A109470 (program): Sum of first n noncubes.
- A109474 (program): a(1)=1, a(2)=3; thereafter, a(n) = least positive integer > a(n-1) and not equal to a(i)+a(j)+a(k) for 1<=i<=j<=k<=n-1.
- A109489 (program): Value of Product[k/sd(k,2),k=1..n], where sd(k,b) is the sum of the digits of k represented in base b.
- A109491 (program): Value of Product_{k=1..n} sigma(k)/sd(k,2), where sd(k,b) is the sum of the digits of k represented in base b.
- A109493 (program): a(n) = 7^((n^2 - n)/2).
- A109498 (program): Number of closed walks of length 2n on the Heawood graph from a given node.
- A109499 (program): Number of closed walks of length n on the complete graph on 5 nodes from a given node.
- A109500 (program): Number of closed walks of length n on the complete graph on 6 nodes from a given node.
- A109501 (program): Number of closed walks of length n on the complete graph on 7 nodes from a given node.
- A109502 (program): Array read by antidiagonals: T(m,n) is the number of closed walks of length n on the complete graph on m nodes, m >= 1, n >= 0.
- A109506 (program): Expansion of (1 - phi(-q)^4)/ 8 in powers of q where phi() is a Ramanujan theta function.
- A109511 (program): Number of subsets of the first n numbers having a common divisor greater than 1.
- A109512 (program): Integers which are not the sum of n and A001462(n).
- A109516 (program): a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,n-1].
- A109517 (program): a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,2(n-1)].
- A109518 (program): a(n)=the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,1;n-1,3(n-1)].
- A109519 (program): a(n) is the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,-1;n-1,n-1].
- A109520 (program): a(n)=the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,-1;n-1,2*(n-1)].
- A109521 (program): a(n)=the (1,2)-entry of the n-th power of the 2 X 2 matrix [0,-1;n-1,3*(n-1)].
- A109522 (program): a(n) = the (1,2)-entry in the matrix P^n + F^n, where the 2 X 2 matrices P and F are defined by P=[0,1;1,0] and F=[0,1;1,1].
- A109523 (program): a(n) is the sum of the (1,2)- and (1,3)-entries of the matrix P^n + T^n, where the 3 X 3 matrices P and T are defined by P = [0,1,0; 0,0,1; 1,0,0] and T = [0,1,0; 0,0,1; 1,1,1].
- A109524 (program): a(n)=the sum of the (1,2)- and (1,3)-entries of the matrix P^n + T^n, where the 3 X 3 matrices P and T are defined by P=[0,1,0;0,0,1;1,0,0] and T=[0,1,0;0,0,1;1,1,0].
- A109534 (program): a(0)=1, a(n)=n+a(n-1) if n mod 2=0, a(n)=3n-a(n-1) if n mod 2 = 1.
- A109535 (program): a(0) = 1, a(n) = n+a(floor(n/2)) if n mod 2 = 0, a(n) = 2n-a(floor((n-1)/2)) if n mod 2 = 1.
- A109536 (program): a(0) = 1, a(n) = n+a(floor(n/2)) if n mod 2 = 0, a(n) = n-a(floor((n-1)/2)) if n mod 2 = 1.
- A109537 (program): a(0)=a(1)=a(2)=a(3)=a(4)=a(5)=1; a(n)=a(n-1)+a(n-2)-a(n-4)+a(n-6) for n>=6.
- A109538 (program): a(n) = a(n-2) + a(n-3) + a(n-4) + a(n-5) + a(n-6) + a(n-7).
- A109539 (program): a(n) = a(n-2)+a(n-3)+a(n-4)+a(n-5)+a(n-6)+a(n-7)+a(n-8).
- A109540 (program): a(n) = a(n-2)+a(n-3)+a(n-4)+a(n-5)+a(n-6)+2*a(n-7)+a(n-8).
- A109541 (program): a(n) = a(n-2)+a(n-3)+a(n-4)+a(n-5)+2*a(n-6)+a(n-7).
- A109543 (program): a(n) = a(n-1) + a(n-3) + a(n-5), with a(1..5) = 1.
- A109544 (program): Expansion of (1 + x - x^3 - 2*x^4)/(1 - x^2 - x^3 - x^4 - x^5).
- A109545 (program): a(n) = 2*a(n-1) + a(n-2) + a(n-3).
- A109546 (program): (4^(n+1)-(-1)^n 9 )/5.
- A109554 (program): pi(n)[prime(n+1)-prime(n)], where pi(n) is the number of prime numbers less than or equal to n and prime(k) is the k-th prime.
- A109555 (program): prime(k) for those k where floor(2*(((prime(k + 1) - prime(k))*PrimePi(k)) mod (8*k)) / k) = m with m = 0.
- A109570 (program): E.g.f.: x/(1-sinh(x)).
- A109572 (program): E.g.f.: x/[1-tan(x)].
- A109573 (program): E.g.f.: 2*x/(1+exp(-2*x)).
- A109576 (program): E.g.f.: x/(1+3x-4x^3)=x/[1-T(3,x)], where T(3,x) is a Chebyshev polynomial.
- A109578 (program): a(n) = (pi(n+1)-pi(n)) * (prime(n+1)-prime(n)), where pi(k) is the number of prime numbers less than or equal to k (= A000720(k)) and prime(k) is the k-th prime number (= A000040(k)).
- A109579 (program): Sum([pi(j+1)-pi(j)][prime(j+1)-prime(j)],j=1..n), where pi(k) is the number of prime numbers less than or equal to k and prime(k) is the k-th prime.
- A109581 (program): E.g.f.: x/(1+x-x^3).
- A109582 (program): Expansion of e.g.f.: -1/(1+x-x^3).
- A109583 (program): Let x^3/(-1-x+x^3)=Sum[b[n]*x^n/n1,{n,0,Infinity}]; a(n) = Abs[b[n]]
- A109584 (program): a(n) = (prime(n+1) - prime(n))^pi(n).
- A109585 (program): a(n) = ( prime(n+1) - prime(n) )^(n+1).
- A109588 (program): n followed by n^2 followed by n^3.
- A109592 (program): Sequence and first differences include all even numbers exactly once and no odd numbers.
- A109594 (program): n followed by n^3 followed by n^2.
- A109595 (program): n^3 followed by n^2 followed by n.
- A109599 (program): a(n) = A070864(n+8) - 4.
- A109603 (program): Numbers n such that 43*n - 41 is prime.
- A109604 (program): Numbers n such that 41*n - 43 is prime.
- A109606 (program): Number of numbers k with 1 < k < n which are relatively prime to n.
- A109607 (program): Sum of coprimes of n greater than 1.
- A109608 (program): Numbers n such that the number of digits required to write the prime factors of n equals the number of digits of n.
- A109609 (program): Expansion of 1/((x-1)*(x+1)*(x^2+x+1)*(x^2+x-1)*(x^2-x+1)*(x^2+1)*(x^4-x^2+1)).
- A109610 (program): Expansion of (1+3*x^4-2*x^7+x^10-x^12)/((x+1)*(x^2+1)*(x^2+x+1)*(x^2-x+1)*(x^4-x^2+1)*(x-1)^2).
- A109612 (program): Numbers n such that the string 44n is prime.
- A109613 (program): Odd numbers repeated.
- A109614 (program): n^3 followed by n followed by n^2.
- A109620 (program): a(n) = (1/3)*n^3 - n^2 - (1/3)*n - 1.
- A109622 (program): Number of different isotemporal classes of diasters with n peripheral edges.
- A109624 (program): Totally multiplicative sequence with a(p) = (p-1)*(p+3) = p^2+2p-3 for prime p.
- A109629 (program): Sequence of Mahler coefficients of the Gray code function.
- A109630 (program): The winning position when playing the “eeny meeny miny moe” game with n players and eliminating every 8th player.
- A109632 (program): In the game of bridge, a(n) is the penalty for going down n tricks in a vulnerable, doubled contract.
- A109633 (program): In the game of bridge, a(n) is the penalty for going down n tricks in a non-vulnerable, doubled contract.
- A109634 (program): Number of 1’s that appear among all ternary strings of length n that contain no consecutive 1’s.
- A109635 (program): Sum of prime(n) and n-th digit of Pi after the decimal point.
- A109637 (program): Numbers n such that the string 55n is prime.
- A109639 (program): Numbers n such that the string 66n is prime.
- A109645 (program): Primes whose decimal expansion has the form ij, where i and j are integers with j < i.
- A109652 (program): a(n) = prime(A000201(n)).
- A109653 (program): Sequence and first differences include all prime numbers exactly once.
- A109664 (program): a(1) = 1; for n>1, a(n) = Sum_{i=1..n-1} a(i)*prime(i).
- A109674 (program): a(n)^(n/a(n)) = A092975(n) and a(n) is a prime.
- A109678 (program): Sequence and first differences include all square numbers exactly once.
- A109680 (program): a(n) = 2^(4n-2) - A104403(n).
- A109681 (program): “Sloping ternary numbers”: write numbers in ternary under each other (right-justified), read diagonals in upward direction, convert to decimal.
- A109683 (program): Ternary equivalents of A109681.
- A109697 (program): Number of partitions of n into parts each equal to 1 mod 5.
- A109698 (program): Number of partitions of n into parts each equal to 2 mod 5.
- A109699 (program): Number of partitions of n into parts each equal to 3 mod 5.
- A109700 (program): Number of partitions of n into parts each equal to 4 mod 5.
- A109701 (program): Number of partitions of n into parts each equal to 1 mod 6.
- A109702 (program): Number of partitions of n into parts each equal to 5 mod 6.
- A109703 (program): Number of partitions of n into parts each equal to 1 mod 7.
- A109704 (program): Number of partitions of n into parts each equal to 2 mod 7.
- A109705 (program): Number of partitions of n into parts each equal to 3 mod 7.
- A109707 (program): Number of partitions of n into parts each equal to 5 mod 7.
- A109712 (program): UnitarySigmaUnitaryPhi(n) or USUP(n).
- A109718 (program): Periodic sequence with period {0,1,0,3}, or n^3 mod 4.
- A109720 (program): Periodic sequence {0,1,1,1,1,1,1} or n^6 mod 7.
- A109722 (program): Sum of first 2n primes.
- A109723 (program): Sum of the first 2n+1 primes.
- A109724 (program): Sum of the first n^2 primes.
- A109725 (program): Divide primes in groups with 2n+1 elements and add together.
- A109731 (program): a(n) = - 4*a(n-2) - a(n-4), a(0) = 1, a(1) = -4, a(2) = -6, a(3) = 15.
- A109742 (program): a(n) = d(n-1) + d(n-2) + (n-1)[d(n-2) + 2d(n-3) + d(n-4)], where d(n), the derangement numbers, are given in A000166. (Let d(n) = 0 if n < 0.)
- A109743 (program): a(2)=1; for n>2, a(n) = A109742(n)/3.
- A109747 (program): E.g.f.: exp(-exp(-x)+1+x).
- A109753 (program): n^3 mod 8; the periodic sequence {0,1,0,3,0,5,0,7}.
- A109754 (program): Matrix defined by: a(i,0) = 0, a(i,j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0; read by ascending antidiagonals.
- A109763 (program): Primes repeated.
- A109764 (program): Sum of the first n^2 squares.
- A109765 (program): Expansion of x/((4*x-1)*(2*x-1)*(x+1)).
- A109767 (program): Triangle T(n,k), 0 <= k <= n, defined by T(n,k) = 2^k*A001497(n,k).
- A109768 (program): a(n) = gcd(3^n-2,2^n-3).
- A109770 (program): Prime(1^2) + prime(2^2) + … + prime(n^2).
- A109771 (program): G.f.: sqrt(1+6*x+x^2).
- A109774 (program): a(n) = (3^(n-1) - 1) * (3^n - 1)/2.
- A109779 (program): a(n) = n! * Sum_{k=1..n} H(k)*(n+1-k)!, where H(k) = Sum_{j=1..k} 1/j.
- A109780 (program): a(n) = n! * Sum_{k=1..n} H(k)*(n-k)!, where H(k) = Sum_{j=1..k} 1/j.
- A109782 (program): Expansion of x*(1+2*x^2-2*x^3+x^4) / ((x-1)*(x^2-2*x-1)*(x^2-x+1)*(x+1)^2).
- A109785 (program): Expansion of (1+x+x^2+x^7+x^8-2*x^10-x^12) / ((x+1)*(x^2+1)*(x^2+x+1)*(x^2-x+1)*(x^4-x^2+1)*(x-1)^2).
- A109786 (program): Expansion of -(x+2*x^2+3*x^3-1+5*x^4)/((x+1)*(x^2-3*x+1)*(1+x^2)).
- A109787 (program): Expansion of -(1-x-2*x^2+11*x^4-3*x^3) / ((x-1)*(x+1)*(x^2-3*x+1)*(1+x^2)).
- A109792 (program): Expansion of e.g.f. log(1+x)/(1-x)^2.
- A109794 (program): a(2n) = A001906(n+1), a(2n+1) = A002878(n).
- A109795 (program): a(n)= n*(1+floor(n/10)).
- A109801 (program): Cumulative sum of squares of primes indexed by squares.
- A109803 (program): Expansion of (x^2+1)*(x+1)^2 / ((x-1)*(x^2+x+1)*(x^2+2*x-1)).
- A109804 (program): Cumulative sum of initial digits of (n base 6).
- A109805 (program): a(n) = prime(n+2)*prime(n+1) - prime(n)*prime(n+1).
- A109808 (program): a(n) = 2*7^(n-1).
- A109810 (program): Number of permutations of the positive divisors of n, where every element is coprime to its adjacent elements.
- A109814 (program): a(n) is the largest k such that n can be written as sum of k consecutive positive integers.
- A109815 (program): n^2 followed by n^3 followed by n.
- A109816 (program): n^2 followed by n followed by n^3.
- A109818 (program): Sum of primes between n and n^2.
- A109819 (program): Product of primes between n and n^2.
- A109823 (program): a(n) is the minimal b >= n such that sum of consecutive integers from n to b is a semiprime.
- A109824 (program): a(n) is the number of consecutive integers starting with n summing up to a semiprime.
- A109825 (program): Initial terms of groups in the partition of the sequence of natural numbers A109411.
- A109826 (program): Final terms of groups in the partition of the sequence of natural numbers A109411.
- A109827 (program): Numbers written in an alternating binary-then-ternary base.
- A109834 (program): Startorial numbers: product of initial digits of integers 1 through n.
- A109841 (program): a(n) is the minimal j >= n such that the sum of consecutive integers from n to j is a palindrome.
- A109842 (program): Number of consecutive integers starting with n needed to sum to a palindrome.
- A109844 (program): a(1) = 1, a(2) = 2, next terms up to a(2n-1) are obtained by multiplying previous terms a(n-1) by n+1, a(n-2) by n+2 etc. a(2) by (2n-2) and a(1) by 2n-1. On similar lines a(2n) = 2n*a(2n-2), a(2n+1) = (2n+1)*a(2n-1) and so on.
- A109845 (program): a(1) = 2; a(2n) = least common multiple of all previous terms + 1. a(2n+1) = least common multiple of all previous terms - 1.
- A109846 (program): Absolute difference between n and its 9’s complement.
- A109847 (program): Least common multiple of n and its 9’s complement.
- A109848 (program): Highest common factor of n and its 9’s complement.
- A109851 (program): a(1) = 1, a(2) = 2; for n > 2, sum of absolute differences of all combinations of pairs of previous terms.
- A109853 (program): a(n) = A109852(2^n).
- A109857 (program): Next 2n-1 odd numbers in decreasing order followed by next 2n even numbers in decreasing order.
- A109866 (program): 9’s complement of the digits of the golden ratio phi (A001622): 9.999999999999… - 1.6180339887… = 8.3819660112501051517954131656334…
- A109868 (program): Numbers which can be differences of successive palindromes in order of their first occurrence.
- A109873 (program): a(n) = product of terms in row n of Pascal’s triangle (A001142) divided by n^k, where n^k is the largest power of n dividing it.
- A109882 (program): Palindromes with either no internal digits or all internal digits are 0.
- A109883 (program): Start subtracting from n its divisors beginning from 1 until one reaches a number smaller than the last divisor subtracted or reaches the last nontrivial divisor < n. Define this to be the perfect deficiency of n. Then a(n) = perfect deficiency of n.
- A109895 (program): Group the natural numbers so that every 2n-th group product is divisible by the single number in the next group. (1), (2,3,4,5), (6), (7,8,9,10,11), (12), (13,14,15,16,17,18,19),(20), (21,22,23,24,25,26,27),(28),… Sequence contains the single members of the odd numbered groups.
- A109896 (program): Group the natural numbers so that every 2n-th group product is divisible by the single number in the next group. (1), (2,3,4,5), (6), (7,8,9,10,11), (12), (13,14,15,16,17,18,19),(20), (21,22,23,24,25,26,27),(28),… Sequence contains the number of terms in the 2n-th group.
- A109900 (program): The (n,r)-th term of the following triangle is T(n)-T(r) for r = 0 to n. The n-th row contains n+1 terms. T(n) = the n-th triangular number = n(n+1)/2. Sequence contains the sum of terms at a 45-degree angle.
- A109901 (program): a(n) = binomial(n^2, n*(n+1)/2).
- A109906 (program): A triangle of coefficients based on A000045 and Pascal’s triangle: t(n,m)=Fibonacci[(n - m + 1)]*Fibonacci[(m + 1)]*Binomial[n, m].
- A109909 (program): a(n) = number of primes of the form k*(n-k)-1.
- A109915 (program): Product of all composite numbers k such that n<k<prime(r) where prime(r-1)<=n, or 1 if this set of k is empty.
- A109916 (program): a(n) = n-th digit after decimal point in e^n.
- A109921 (program): a(2n) = prime(n). a(2n+1) = sum of composite numbers between prime(n) and prime(n+1). We define a(1) = 1.
- A109922 (program): a(n) = floor(lcm(1,2,…n)/(1+2+…+n)).
- A109923 (program): a(n) = lcm(1,2,3,…,prime(n))/(1 + 2 + … + prime(n)).
- A109925 (program): Number of primes of the form n - 2^k.
- A109932 (program): a(n) = f^n(n) = f applied n times to n, where f(n) = A007425(n).
- A109934 (program): Composite numbers which are not the sum of two distinct primes.
- A109940 (program): Largest k-digit multiple of n where k is the number of digits in n.
- A109952 (program): Degrees Centigrade for which Fahrenheit is a prime.
- A109954 (program): Riordan array (1/(1+x)^3,x/(1+x)^2).
- A109955 (program): Number triangle binomial(n+2k,3k).
- A109956 (program): Inverse of Riordan array (1/(1-x), x/(1-x)^3), A109955.
- A109960 (program): Number triangle binomial(n+3k,4k).
- A109961 (program): Expansion of (1-x)^3/(1-4x+5x^2-4x^3+x^4).
- A109962 (program): Inverse of Riordan array (1/(1-x), x/(1-x)^4), A109960.
- A109964 (program): a(n) = floor(sqrt(Sum_{i<n} a(i))), with a(0)=1.
- A109965 (program): Sum_i {i<n} floor(sqrt(a(i))) with a(0) = 1.
- A109966 (program): a(n) = 8^((n^2-n)/2).
- A109970 (program): Riordan array (1,x(1-x)^2).
- A109971 (program): Inverse of Riordan array (1,x(1-x)^2), A109970.
- A109975 (program): Second differences of A045623, prefixed by an initial 1.
- A109980 (program): Number of Delannoy paths of length n with no (1,1)-steps on the line y=x.
- A109983 (program): Triangle read by rows: T(n, k) (0<=k<=2n) is the number of Delannoy paths of length n, having k steps.
- A109984 (program): a(n) = number of steps in all Delannoy paths of length n.
- A109990 (program): Numbers n such that the string 77n is prime.
- A109991 (program): Numbers k such that the string 88k is prime.
- A109992 (program): Numbers n such that the string 99n is prime.
- A109995 (program): Number of unlabeled ordered minimal T_0-covers of an n-set, cf. A094545.
- A109999 (program): Integer part of Lorentz gamma factor = 1/sqrt(1 - (beta)^2) for beta = 0.9999…(with 9 appearing n times) = 1 - 10^(-n).
- A110001 (program): n followed by n^2 followed by n^3 followed by n^4.
- A110003 (program): n followed by n^3 followed by n^2 followed by n^4.
- A110004 (program): n followed by n^3 followed by n^4 followed by n^2.
- A110005 (program): n followed by n^2 followed by n^4 followed by n^3.
- A110006 (program): a(n)=n-floor(phi*floor(phi^-1*floor(phi*floor(phi^-1*n)))) where phi=(1+sqrt(5))/2.
- A110007 (program): a(n)=n-floor(phi*floor(phi^-1*floor(phi*floor(phi^-1*floor(phi*floor(phi^-1*n)))))) where phi=(1+sqrt(5))/2.
- A110008 (program): n followed by n^4 followed by n^3 followed by n^2.
- A110009 (program): n followed by n^4 followed by n^2 followed by n^3.
- A110010 (program): a(n)=n-F(F(F(F(n)))) where F(x)=floor(phi*floor(x/phi)) and phi=(1+sqrt(5))/2.
- A110011 (program): a(n)=n-F(F(F(F(F(n)))))=n-F^5(n) where F(x)=floor(phi*floor(x/phi)) and phi=(1+sqrt(5))/2.
- A110012 (program): a(n) = n - F(F(n)) where F(x)=floor(sqrt(2)*floor(x/sqrt(2)).
- A110013 (program): Squares of the form 4p + 5, where p is a prime.
- A110014 (program): Primes p such that 6p + 7 is a square.
- A110015 (program): Squares of the form 6p + 7, where p is a prime.
- A110016 (program): Numbers n such that (n^2-7)/6 is prime.
- A110026 (program): Minimal number of times a rectangular grid of n X n+1 elements can be slid along a 45-degree line before a rotated version of the initial grid appears.
- A110034 (program): Row sums of a characteristic triangle for the Fibonacci numbers.
- A110035 (program): Row sums of an unsigned characteristic triangle for the Fibonacci numbers.
- A110036 (program): Constant terms of the partial quotients of the continued fraction expansion of 1 + Sum_{n>=0} 1/x^(2^n), where each partial quotient has the form {x + a(n)} after the initial constant term of 1.
- A110037 (program): Signed version of A090678 and congruent to A088567 mod 2.
- A110043 (program): a(0) = 1, a(1) = 2; for n>1, a(n) = n*a(n-1) + (-1)^n.
- A110044 (program): a(0) = 11, a(1) = 23; for n > 1, a(n) = |a(n-1) - a(n-2)|.
- A110046 (program): Expansion of (1+4*x-12*x^2-16*x^3)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)).
- A110047 (program): Expansion of (1+2*x-4*x^2)/((2*x+1)*(2*x-1)*(4*x^2+4*x-1)).
- A110048 (program): Expansion of 1/((1+2*x)*(1-4*x-4*x^2)).
- A110050 (program): Expansion of (2+9*x-24*x^3+16*x^4-30*x^2) / ((1-x)*(2*x+1)*(2*x-1)*(4*x^2+4*x-1)).
- A110051 (program): Expansion of (1-x+2*x^3+x^2)/((1-x)*(2*x+1)*(2*x-1)*(4*x^2+4*x-1)).
- A110052 (program): Expansion of x*(-1+4*x)/((x-1)*(2*x-1)*(4*x^2+4*x-1)).
- A110061 (program): Expansion of x^2*(-3+4*x)/(1-x^3+x^4).
- A110062 (program): Expansion of (1 - x + 2*x^2) / (1 - x^3 + x^4).
- A110063 (program): Expansion of -(3 - x + 2*x^2) / (1 - x^3 + x^4).
- A110064 (program): a(n+4) = a(n+1) - a(n), a(0) = 1, a(1) = -4, a(2) = 0, a(3) = 1.
- A110083 (program): a(n+1) = Sum_{k=0..n} (n!/k!)*binomial(n,k)*a(k).
- A110088 (program): tau(n)^omega(n), where tau=A000005 and omega=A001221.
- A110090 (program): Numerators of sequence of rationals defined by r(n) = n for n<=1 and for n>1: r(n) = (sum of denominators of r(n-1) and r(n-2))/(sum of numerators of r(n-1) and r(n-2)).
- A110091 (program): Denominators of sequence of rationals defined by r(n) = n for n<=1 and for n>1: r(n) = (sum of denominators of r(n-1) and r(n-2))/(sum of numerators of r(n-1) and r(n-2)).
- A110098 (program): Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n, having k return steps to the line y = x from the line y = x+1 (i.e., E steps from the line y=x+1 to the line y = x).
- A110099 (program): Number of return steps to the line y = x from the line y = x+1 (i.e., E steps from the line y = x+1 to the line y = x) in all Delannoy paths of length n.
- A110110 (program): Number of symmetric Schroeder paths of length 2n (A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis).
- A110111 (program): Sequence associated to the recurrence b(n) = b(n-1) + 3*b(n-2).
- A110113 (program): Diagonal sums of A083856.
- A110117 (program): a(n) = floor(n * (sqrt(2) + sqrt(3))).
- A110118 (program): a(n) = floor(n*(sqrt(6) + sqrt(2) + 2)/4).
- A110122 (program): Number of Delannoy paths of length n with no EE’s crossing the line y = x (i.e., no two consecutive E steps from the line y = x+1 to the line y = x-1).
- A110127 (program): Number of EE’s crossing the line y = x (i.e., two consecutive E steps from the line y = x+1 to the line y = x-1) in all Delannoy paths of length n.
- A110129 (program): Central coefficients of a scaled Legendre triangle.
- A110131 (program): Determinant of n X n matrix M_{i,j} = 2^i*P_i(j), where P_i(j) is the Legendre polynomial of order i at j and i and j are 0-based.
- A110132 (program): a(n) = floor(n/2)^ceiling(n/2).
- A110133 (program): Numbers which are the sides that belong to only one primitive Pythagorean triangle.
- A110138 (program): a(n) = ceiling(n/2)^floor(n/2).
- A110139 (program): Floor(n/2)^floor(n/2).
- A110140 (program): Binomial transform of n^n (with interpolated zeros).
- A110144 (program): Terms of A110142 at positions p(n)+1, where p(n) = A000041(n) is the number of partitions of n; a(n) = A110142(p(n)+1) for n>=1, with a(0) = 1.
- A110145 (program): a(n) = Sum_{k=0..n} C(n,k)^2*mod(k,2).
- A110146 (program): n^(n+1) mod n+2.
- A110147 (program): 10^((n^2-n)/2).
- A110149 (program): a(0) = 1, a(1) = 3; for n>1, a(n) = n*a(n-1) + (-1)^n.
- A110152 (program): G.f.: A(x) = Product_{n>=1} 1/(1 - 2^n*x^n)^(2/2^n).
- A110153 (program): G.f.: A(x) = Product_{n>=1} 1/(1 - 3^n*x^n)^(3/3^n).
- A110157 (program): a(n) = a(rad(n) - 1) + 1, where rad(n) is the squarefree kernel of n, rad=A007947.
- A110158 (program): Expansion of x^4 / ((x+1)*(2*x^3-2*x^2-2*x+1)*(x-1)^2).
- A110159 (program): a(n) = (n+1)(n+2)(n+3)(9n^2 + 26n + 20)/120.
- A110161 (program): Expansion of x(1-x^2)/(1-x^2+x^4).
- A110162 (program): Riordan array ((1-x)/(1+x), x/(1+x)^2).
- A110164 (program): Expansion of (1-x^2)/(1+2x).
- A110165 (program): Riordan array (1/sqrt(1-6x+5x^2),(1-3x-sqrt(1-6x+5x^2))/(2x).
- A110166 (program): Row sums of Riordan array A110165.
- A110167 (program): Diagonal sums of Riordan array A110165.
- A110168 (program): Riordan array ((1-x^2)/(1+3x+x^2),x/(1+3x+x^2)).
- A110169 (program): Triangle read by rows: T(n,k) (0<=k<=n) is the number of Delannoy paths of length n that start with exactly k (1,1) steps.
- A110170 (program): First differences of the central Delannoy numbers (A001850).
- A110171 (program): Triangle read by rows: T(n,k) (0 <= k <= n) is the number of Delannoy paths of length n that start with exactly k (0,1) steps (or, equivalently, with exactly k (1,0) steps).
- A110180 (program): Triangle of generalized central trinomial coefficients.
- A110181 (program): Row sums of number triangle A110180.
- A110184 (program): Number of (1,1)-steps on the lines y=x, y=x+1 and y=x-1 in all Delannoy paths of length n.
- A110185 (program): Coefficients of x in the partial quotients of the continued fraction expansion exp(1/x) = [1, x - 1/2, 12*x, 5*x, 28*x, 9*x, 44*x, 13*x, …]. The partial quotients all have the form a(n)*x except the constant term of 1 and the initial partial quotient which equals (x - 1/2).
- A110190 (program): Number of (1,0)-steps on the lines y=0 and y=1 in all Schroeder paths of length 2n (a Schroeder path of length 2n is a path from (0,0) to (2n,0), consisting of steps U=(1,1), D=(1,-1) and H=(2,0) and never going below the x-axis).
- A110191 (program): Decimal expansion of 1/6 - 1/(2*Pi).
- A110195 (program): a(n) = 11^((n^2-n)/2).
- A110197 (program): Number triangle of sums of squared binomial coefficients.
- A110198 (program): Diagonal sums of number triangle A110197.
- A110199 (program): a(n) = Sum_{k=0..floor(n/2)} Catalan(k).
- A110202 (program): a(n) = sum of squares of numbers < 2^n having exactly 2 ones in their binary representation.
- A110206 (program): Row sums of triangle A110205, where A110205(n,k) equals the sum of cubes of numbers < 2^n having exactly k ones in their binary expansion.
- A110208 (program): 1 + sum of first n semiprimes.
- A110209 (program): 1 + sum of first n 3-almost primes.
- A110210 (program): a(n+3) = 6*a(n) - 5*a(n+2), a(0) = -1, a(1) = 1, a(2) = -5.
- A110211 (program): a(n+3) = 6*a(n) - 5*a(n+2), a(0) = -1, a(1) = 3, a(2) = -15.
- A110212 (program): a(n+3) = 6*a(n) - 5*a(n+2), a(0) = -1, a(1) = 5, a(2) = -25.
- A110213 (program): a(n+3) = 6*a(n) - 5*a(n+2), a(0) = 1, a(1) = -7, a(2) = 35.
- A110224 (program): a(n) = Fibonacci(n)^3 + Fibonacci(n+1)^3.
- A110226 (program): 1 + sum of first n 4-almost primes.
- A110236 (program): Number of (1,0) steps in all peakless Motzkin paths of length n (can be easily translated into RNA secondary structure terminology).
- A110239 (program): Number of (1,1) steps in all peakless Motzkin paths of length n.
- A110240 (program): Decimal form of binary integer produced by the ON cells at n-th generation following Wolfram’s Rule 30 cellular automaton starting from a single ON-cell represented as 1.
- A110241 (program): J(n)^3+J(n+1)^3, where J(n) = the Jacobsthal number A001045(n).
- A110254 (program): Square-indexed values of A110243.
- A110256 (program): Denominators in the fractional coefficients that form the partial quotients of the continued fraction representation of the inverse tangent of 1/x.
- A110257 (program): Numerators in the coefficients that form the odd-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x.
- A110258 (program): Denominators in the coefficients that form the odd-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x.
- A110260 (program): Denominators in the coefficients that form the even-indexed partial quotients of the continued fraction representation of the inverse tangent of 1/x.
- A110266 (program): Number of blocks of ON cells in n-th row of triangle generated by Wolfram’s “Rule 30”.
- A110267 (program): Total number of black cells at the first n generations of a single black cell following Wolfram’s Rule 30 cellular automaton.
- A110269 (program): n mod 2 + n mod 3.
- A110270 (program): a(n) = (n mod 2)*(n mod 3).
- A110271 (program): Inverse of Riordan array (1/(1-x)^2,x(1-x)/(1+x)), A104698.
- A110272 (program): a(n) = Pell(n)^3.
- A110273 (program): a(n) = Pell(n)^3 + Pell(n+1)^3.
- A110277 (program): Values of n such that the perfect deficiency (A109883) of n is a square.
- A110284 (program): Squares of the form 4p - 3, where p is a prime.
- A110286 (program): a(n) = 15*2^n.
- A110287 (program): 17*2^n.
- A110288 (program): 19*2^n.
- A110291 (program): Riordan array (1/(1-x),x(1+2x)).
- A110293 (program): a(2*n) = A001570(n), a(2*n+1) = A011943(n+1).
- A110294 (program): a(2*n) = A028230(n), a(2*n+1) = -A067900(n+1).
- A110295 (program): a(n) = prime(n)*2^(n-1).
- A110299 (program): a(n) = Sum_{i=0..n-1} 2^i*prime(n-i).
- A110301 (program): Integers written in base “triangle”.
- A110303 (program): Alternators.
- A110307 (program): Expansion of (1+2*x)/((x^2+x+1)*(x^2+5*x+1)).
- A110308 (program): Expansion of -x*(2+x)/((x^2+x+1)*(x^2+5*x+1)).
- A110309 (program): Expansion of (1+3*x+x^2)/((x^2+5*x+1)*(x^2+x+1)).
- A110310 (program): Expansion of (1-x+x^2)/((x^2+x+1)*(x^2+5*x+1)).
- A110311 (program): Expansion of 1 / ((x^2+5*x+1)*(x^2+x+1)).
- A110313 (program): Expansion of e.g.f. exp(x)/(1-x-x^2).
- A110314 (program): Inverse of number triangle related to Fibonacci numbers.
- A110315 (program): Diagonal sums of the Fibonacci related number triangle A110314.
- A110316 (program): a(n) is the number of different shapes of balanced binary trees with n nodes. The tree is balanced if the total number of nodes in the left and right branch of every node differ by at most one.
- A110318 (program): Number of arcs covered by other arcs in all RNA secondary structures of size n+5 (i.e., with n+5 nodes).
- A110319 (program): Triangle read by rows: T(n,k) (1 <= k <= n) is number of RNA secondary structures of size n (i.e., with n nodes) having k blocks (an RNA secondary structure can be viewed as a restricted noncrossing partition).
- A110320 (program): Number of blocks in all RNA secondary structures with n nodes (an RNA secondary structure can be viewed as a restricted noncrossing partition).
- A110321 (program): A Jacobsthal number related number triangle.
- A110322 (program): Row sums of a number triangle related to the Jacobsthal numbers.
- A110324 (program): Inverse of a number triangle related to the Jacobsthal numbers.
- A110325 (program): Row sums of number triangle related to the Jacobsthal numbers.
- A110326 (program): Diagonal sums of triangle A110324.
- A110327 (program): Triangle read by rows: T(n,k) = n!*Pell(n-k+1)/k!, where Pell(n)=A000129(n).
- A110328 (program): Row sums of a number triangle related to the Pell numbers.
- A110330 (program): Inverse of a number triangle related to the Pell numbers.
- A110331 (program): Row sums of a number triangle related to the Pell numbers.
- A110332 (program): Diagonal sums of number a triangle related to the Pell numbers.
- A110344 (program): a(n) = Sum_{k=0..n-1} (n+k) = n(3n-1)/2 if n is even; a(n) = Sum_{k=0..n-1} (n-k) = n(n+1)/2 if n is odd.
- A110345 (program): a(n) = n + (n+1) + (n+2) + … n terms if n is odd, else a(n) = n + (n-1) + (n-2) + … n terms = n(n+1)/2 = n-th triangular number if n is even.
- A110346 (program): Largest multiple of n in n + (n-1) +(n-2) + … (n-k).
- A110347 (program): a(n) = meantorial(n) = the product of the set of n closest numbers with an arithmetic mean of n.
- A110348 (program): a(2) = 1 by definition; otherwise a(n) = A109347(n)/n.
- A110349 (program): a(n) = n + (n+1) + (n-1) + (n+2) + (n-2) … n terms.
- A110350 (program): Least sum (n+1) + (n+2) + …+(n+k) >= (n(n+1)/2), the n-th triangular number.
- A110356 (program): Array read by antidiagonals: T(n,k) (n>=3, k>=3) = minimal number of polygonal pieces in a dissection of a regular n-gon to a regular k-gon (conjectured).
- A110357 (program): Least integer of the form n*(n+k)/(n-k).
- A110359 (program): a(n) = n+1 if n+1 is a prime else a(n) = 2n+1 if 2n+1 is a prime else a(n) = 2*(2n+1) +1 =g(n) if this number is prime else the next candidate is 2* (g(n) +1 etc.
- A110365 (program): a(1)=2, a(n+1) = a(n)*A010888(a(n)).
- A110369 (program): (Digit 1 repeated n times) + n.
- A110370 (program): Floor[ (digits n times n) divided by digits (n times 1)].
- A110371 (program): a(n)=[(n+1)(n+2)(n+3)…(2n)]/(1+2+3+…+n).
- A110372 (program): a(n) = F(n+1)!/F(n)! where F(n) = n-th Fibonacci number.
- A110373 (program): a(n) = Sum_{prime p <= n} n!/p.
- A110376 (program): a(n) = Sum_{r < n, gcd(r,n)=1} n!/r.
- A110377 (program): a(n) = Sum_{r < n, gcd(r,n)=1} n!/r!.
- A110378 (program): a(n) = Sum_{prime p <= n} n!/p!.
- A110379 (program): a(n) = Sum_{composite c <= n} n!/c!.
- A110380 (program): a(n) = min{p + q + r + …} where p,q,r,… are distinct unary numbers - containing only ones, i.e., of the form (10^k - 1)/9 - formed by using a total of n ones.
- A110382 (program): Numbers which are sum of distinct unary numbers (containing only ones), i.e., numbers which are sum of distinct numbers of the form (10^k - 1)/9.
- A110388 (program): a(n) = F(n)*F(n+1) mod 9, where F(n) = n-th Fibonacci number.
- A110389 (program): Integers with mutual residues -1.
- A110391 (program): a(n) = L(3*n)/L(n), where L(n) = Lucas number.
- A110396 (program): 10’s complement factorial of n: a(n) = (10’s complement of n)*(10’s complement of n-1)*…*(10’s complement of 2)*(10’s complement of 1).
- A110397 (program): a(n) = (10^k - n)(10^k + n), where k is the number of digits in n.
- A110399 (program): Expansion of (theta_3(q)*theta_3(q^7) - 1)/2 in powers of q.
- A110412 (program): Sum_{d<n is a divisor of n} tau(n-d).
- A110414 (program): n! concatenated with n divided by n.
- A110415 (program): a(n) = n concatenated with n! divided by n.
- A110422 (program): a(n) = sum( (-1)^(r+1)*(n-r)*r, r = 1..floor(n/2) ).
- A110425 (program): The r-th term of the n-th row of the following array contains the sum of r successively decreasing integers beginning from n. 0<r<=n. e.g. the row corresponding to 4 contains 4, (3+2),{(1) +(0)+(-1)}, {(-2)+(-3)+(-4)+(-5)} —-> 4,5,0,-14 1 2 1 3 3 -3 4 5 0 -14 5 7 3 -10 -35 6 9 6 -6 -30 -69 … Sequence contains the array by rows.
- A110426 (program): The r-th term of the n-th row of the following array contains the sum of r successively decreasing integers beginning from n. 0<r<=n. e.g. the row corresponding to 4 contains 4, (3+2),{(1) +(0)+(-1)}, {(-2)+(-3)+(-4)+(-5)} —-> 4,5,0,-14 1 2 1 3 3 -3 4 5 0 -14 5 7 3 -10 -35 6 9 6 -6 -30 -69 … Sequence contains the row sums.
- A110427 (program): The r-th term of the n-th row of the following array contains the sum of r successively decreasing integers beginning from n. 0 < r <= n. Sequence contains the leading diagonal.
- A110428 (program): a(1) = 1 and a(2) = 2. Subsequent terms are generated like this: if a(m) is the last term available – say a(2) – then a(m+1) = a(m) * a(m-1), a(m+2) = a(m) * a(m-1) * a(m-2), …, a(2*m-1) = a(m) * a(m-1) * a(m-2) * … * a(2) * a(1), a(2*m) = a(2*m-1) * a(2*m-2), and so on.
- A110430 (program): Arithmetic mean of all n-digit positive even numbers.
- A110431 (program): Average of positive multiples of 3 with n decimal digits, rounded down.
- A110436 (program): A weighted sum of Jacobi function values.
- A110437 (program): A weighted sum of Jacobi function values.
- A110440 (program): Triangular array formed by the little Schröder numbers s(n,k).
- A110441 (program): Triangular array formed by the Mersenne numbers.
- A110444 (program): Binary expansion of A074988.
- A110446 (program): Triangle of Delannoy paths counted by number of diagonal steps not preceded by an east step.
- A110448 (program): G.f.: A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), where A056045(n) = Sum_{d|n} binomial(n,d).
- A110449 (program): Triangle read by rows: T(n,k) = n*((2*k+1)*n+1)/2, 0<=k<=n.
- A110450 (program): a(n) = n*(n+1)*(n^2+n+1)/2.
- A110451 (program): a(n) = n*(4*n^2 + 2*n + 1).
- A110467 (program): Convolution of 4^n*n! and n!.
- A110468 (program): a(n) = (2*n + 1)!/(n + 1).
- A110469 (program): Convolution of J(n)*n! and n! where J(n)=A001045(n), n-th Jacobsthal number.
- A110471 (program): Prime analog of Baum-Sweet sequence: a(n) = 1 if binary representation of n contains no block of consecutive zeros of exactly prime length; otherwise a(n) = 0.
- A110473 (program): Integers not in “array with primes”.
- A110475 (program): Number of symbols ‘*’ and ‘^’ to write the canonical prime factorization of n.
- A110477 (program): a(n) = Sum_{k=1..n} k*(prime(k) - k).
- A110480 (program): Numbers n such that (n^2+6)/5 is prime.
- A110481 (program): Squares of the form 5p - 6, where p is prime.
- A110482 (program): Primes p such that 5*p - 6 is square.
- A110484 (program): Squares of the form p*q + p + q + 2, where p and q are primes.
- A110485 (program): n^2 followed by n followed by n^4 followed by n^3.
- A110491 (program): Expansion of e.g.f.: sqrt(1+2x)/sqrt(1-2x).
- A110494 (program): Least k such that prime(n)^2 divides binomial(2k,k).
- A110496 (program): Least k such that prime(n)^3 divides binomial(2k,k).
- A110497 (program): a(1) = 1; a(m) = maximum denominator possible with a continued fraction [b(1);b(2),b(3),…,b(m-1)], where (b(1),b(2),b(3),…,b(m-1)) is a permutation of (a(1),a(2),a(3),…,a(m-1)).
- A110501 (program): Unsigned Genocchi numbers (of first kind) of even index.
- A110503 (program): Triangle, read by rows, which shifts one column left under matrix inverse.
- A110505 (program): Numerators of unsigned columns of triangle A110504: a(n) = n!*A110504(n,0) = (-1)^k*n!*A110504(n+k,k) for all k >= 0.
- A110506 (program): Riordan array (1/(1-xc(2x)),xc(2x)/(1-xc(2x))), c(x) the g.f. of A000108.
- A110507 (program): Number of nodes in the smallest cubic graph with crossing number n.
- A110509 (program): Riordan array (1, x(1-2x)).
- A110510 (program): Riordan array (1, x*c(2x)), c(x) the g.f. of A000108.
- A110511 (program): Riordan array (1/(1+x), x(1-x)/(1+x)^2).
- A110512 (program): Expansion of (1 + x)/(1 + x + 2x^2).
- A110513 (program): Expansion of (1 + x)/(1 + 2x + x^3).
- A110514 (program): Expansion of (1 - x + x^2 + x^3)/(1 - x^2 - x^4 + x^6).
- A110515 (program): Sequence array for (1 - x + x^2 + x^3)/(1 - x^4).
- A110516 (program): Expansion of (1-x+x^2+x^3)/(1+x-x^4-x^5).
- A110517 (program): Riordan array (1,x(1-3x)).
- A110518 (program): Riordan array (1, x*c(3x)), c(x) the g.f. of A000108.
- A110519 (program): Riordan array (1/(1-xc(3x)), xc(3x)/(1-xc(3x))), c(x) the g.f. of A000108.
- A110520 (program): Expansion of 1/(1-2*x*c(3*x)), c(x) the g.f. of A000108.
- A110522 (program): Riordan array (1/(1+x), x(1-2x)/(1+x)^2).
- A110523 (program): Expansion of (1 + x)/(1 + x + 3*x^2).
- A110524 (program): Expansion of (1 + x)/(1 + 2*x + 2*x^3).
- A110525 (program): Expansion of 1/(1-x^2*c(3x)), c(x) the g.f. A000108.
- A110526 (program): a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 3.
- A110527 (program): a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 0, a(1) = 1, a(2) = 8.
- A110528 (program): a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 1, a(1) = 10, a(2) = 37.
- A110532 (program): a(n) = floor(n/2) + floor(n/5).
- A110533 (program): a(n) = floor(n/2) * floor(n/5).
- A110547 (program): Number of sides of regular polygons whose interior angles (in degrees) are not integers.
- A110548 (program): One of the three ordered sets of positive integers that solves the minimal magic die puzzle.
- A110549 (program): Period 8: repeat [1, 2, 4, 3, 3, 4, 2, 1].
- A110550 (program): Periodic {1,3,2,4,4,2,3,1}.
- A110551 (program): Period 6: repeat [1, 3, 5, 5, 3, 1].
- A110552 (program): A triangular array related to A077028 and distributing the values of A007582.
- A110555 (program): Triangle of partial sums of alternating binomial coefficients: T(n,k) = Sum_{j=0..k} binomial(n,j)*(-1)^j; n >= 0, 0 <= k <= n.
- A110556 (program): a(n) = binomial(2*n-1,n)*(-1)^n for n>0; a(0) = 1.
- A110558 (program): Numbers n such that (n^2-8)/8 is prime.
- A110559 (program): Least j such that j*n^2 -1 and j*n^2 +1 are twin primes.
- A110560 (program): Numerators of T(n+1)/n! reduced to lowest terms, where T(n) are the triangular numbers A000217.
- A110561 (program): Denominators of T(n+1)/n! reduced to lowest terms, where T(n) are the triangular numbers A000217.
- A110562 (program): Numbers n such that n in binary representation has a block of exactly a nontrivial pentagonal number of zeros.
- A110565 (program): Results from a change in the rules leading to sequence A097357.
- A110566 (program): a(n) = lcm{1,2,…,n}/denominator of harmonic number H(n).
- A110567 (program): a(n) = n^(n+1) + 1.
- A110568 (program): Period 6: repeat [1, 0, 2, 2, 0, 1].
- A110569 (program): Period 6: repeat [2, 1, 3, 3, 1, 2].
- A110571 (program): Sums of rows of the triangle in A110570.
- A110573 (program): Numbers n such that the string 666n is prime.
- A110574 (program): Binary strings that have 1’s where ‘odious numbers’ occur, 0’s elsewhere and every term ends with the n-th odious number index.
- A110586 (program): Squares of the form 6p+7 for p prime (A110015) that are squares of a prime.
- A110587 (program): Primes p such that 6q+7=p^2, q prime.
- A110588 (program): Squares of the form 2*p+3 that are squares of primes.
- A110589 (program): Primes p such that 2*q+3 = p^2, where q is prime.
- A110591 (program): Number of digits in base-4 representation of n.
- A110592 (program): Number of digits in base-5 representation of n. String length of A007091.
- A110593 (program): a(1) = 3, a(n+1) = 2*(3^n).
- A110594 (program): a(1) = 4, a(2) = 12, for n>1: a(n) = 3*4^(n-1).
- A110595 (program): a(1)=5. For n > 1, a(n) = 4*5^(n-1) = A005054(n).
- A110601 (program): a(n) = phi(n)*tau(n)^2, where phi is Euler’s totient function and tau(n) is the number of divisors of n.
- A110603 (program): Numbers n whose base 5 representations, interpreted as base 10 integers, are semiprimes.
- A110604 (program): Numbers n whose base 6 representations, interpreted as base 10 integers, are semiprimes.
- A110605 (program): Numbers n whose base 7 representations, interpreted as base 10 integers, are semiprimes.
- A110606 (program): Numbers n whose base 8 representations, interpreted as base 10 integers, are semiprimes.
- A110607 (program): Numbers n whose base 9 representations, interpreted as base 10 integers, are semiprimes.
- A110608 (program): Number triangle T(n,k) = binomial(n,k)*binomial(2n,n-k).
- A110609 (program): a(n) = n * C(2*n,n-1).
- A110610 (program): Maximal value of sum(p(i)p(i+1),i=1..n), where p(n+1)=p(1), as p ranges over all permutations of {1,2,…,n}.
- A110611 (program): Minimal value of sum(p(i)p(i+1),i=1..n), where p(n+1)=p(1), as p ranges over all permutations of {1,2,…,n}.
- A110613 (program): a(n+3) = 5*a(n+2) - 2*a(n+1) - 8*a(n), a(0) = 1, a(1) = 0, a(2) = 3.
- A110614 (program): a(n+3) = 5*a(n+2) - 2*a(n+1) - 8*a(n), a(0) = 1, a(1) = 5, a(2) = 15.
- A110616 (program): A convolution triangle of numbers based on A001764.
- A110617 (program): The decimal expansion of 1/64532 (related to an optimal mixed strategy for Hofstadter’s million dollar game).
- A110622 (program): n^2 followed by n followed by n^3 followed by n^4.
- A110650 (program): n^2 followed by n^4 followed by n followed by n^3.
- A110651 (program): n^2 followed by n^4 followed by n^3 followed by n.
- A110652 (program): n^2 followed by n^3 followed by n^4 followed by n.
- A110653 (program): n^2 followed by n^3 followed by n followed by n^4.
- A110654 (program): a(n) = ceiling(n/2), or: a(2*k) = k, a(2*k+1) = k+1.
- A110655 (program): a(n) = A110654(A110654(n)).
- A110656 (program): a(n) = A110654(A110654(A110654(n))).
- A110657 (program): a(n) = A028242(A028242(n)).
- A110658 (program): a(n) = A028242(A028242(A028242(n))).
- A110659 (program): a(n) = A028242(A110654(n)).
- A110660 (program): Oblong (promic) numbers repeated.
- A110661 (program): Triangle read by rows: T(n,k) = total number of divisors of k, k+1, …, n (1 <= k <= n).
- A110662 (program): Triangle read by rows: T(n,k) is the sum of the sums of divisors of k, k+1, …, n (1 <= k <= n).
- A110663 (program): Triangle read by rows: T(n,k) = Sum_{j=k..n} phi(j) (1<=k<=n), where phi is Euler’s totient function.
- A110664 (program): Triangle read by rows: T(n,k)=sum(bigomega(j),j=k..n) (1<=k<=n), where bigomega(j) is the number of prime divisors of j, counted with multiplicities.
- A110665 (program): Sequence is {a(0,n)}, where a(m,0)=0, a(m,n) = a(m-1,n)+a(m,n-1) and a(0,n) is such that a(n,n) = n for all n.
- A110666 (program): Sequence is {a(1,n)}, where a(m,n) is defined at sequence A110665.
- A110667 (program): Sequence is {a(2,n)}, where a(m,n) is defined at sequence A110665.
- A110668 (program): Sequence is {a(3,n)}, where a(m,n) is defined at sequence A110665.
- A110669 (program): Sequence is {a(4,n)}, where a(m,n) is defined at sequence A110665.
- A110670 (program): Sequence is {a(5,n)}, where a(m,n) is defined at sequence A110665.
- A110671 (program): Sequence is {a(6,n)}, where a(m,n) is defined at sequence A110665.
- A110672 (program): Sequence is {a(7,n)}, where a(m,n) is defined in sequence A110665.
- A110673 (program): Numbers that are neither the sum nor the difference of two primes.
- A110678 (program): a(n) = -n^2 - n + 72.
- A110679 (program): a(n+3) = 3*a(n+2) + 5*a(n+1) + a(n), a(0) = 1, a(1) = 2, a(2) = 11.
- A110681 (program): A convolution triangle of numbers based on A071356.
- A110691 (program): Kekulé numbers for certain benzenoids.
- A110695 (program): Kekulé numbers for certain benzenoids of trigonal symmetry.
- A110696 (program): Kekulé numbers for certain benzenoids of trigonal symmetry.
- A110697 (program): Kekulé numbers for certain benzenoids of trigonal symmetry.
- A110707 (program): Number of linear arrangements of n blue, n red and n green items such that there are no adjacent items of the same color (first and last elements considered as adjacent).
- A110728 (program): Digital factorial: a(0) = 1, a(n) = n * (the sum of the digits of a(n-1)).
- A110729 (program): Factorial terms of Digital factorial (A110728).
- A110730 (program): Irregular triangle read by rows in which row n lists n 1’s followed by (n-1) 2’s followed by (n-3) 3’s … followed by 1 n.
- A110737 (program): Row sums in A112668.
- A110738 (program): a(n) = common ratio for row n in A112668.
- A110739 (program): Arithmetic mean of row n in A112668.
- A110748 (program): Form triangle shown below, in which the n-th row contains n terms of an arithmetic progression with first term 1 and common difference n. Then a(n) = terms of the n-th row (mod 10), concatenated.
- A110749 (program): Triangle read by rows with the n-th row containing the first n multiples of n with digits reversed.
- A110751 (program): Numbers n such that n and its digital reversal have the same prime divisors.
- A110765 (program): Let n in binary be a k-digit number say abbaaa… where a = 1 and b = 0. a(n) = 2^a*3^b*5^b*7*a… primes in increasing order raised to the powers starting from the MSB.
- A110766 (program): Fractalization of Pi.
- A110769 (program): The r-th term of the n-th row of the following triangle contains sum of r successive numbers in decreasing order beginning from T(n)-T(r-1) where T(n) is the n-th triangular number. 1 3 3 6 9 6 10 17 18 10 15 27 33 30 15 … Sequence contains the triangle by rows.
- A110770 (program): Triangle read by rows: T(n,k) = binomial(t(n) - t(k-1),k), where t(j) = j*(j+1)/2; 1<=k<=n.
- A110771 (program): The r-th term of the n-th row of the following triangle is C[{T(n)-T(r-1)},r] where T(n) is the n-th triangular number. 1 3 1 6 10 1 10 36 35 1 … Sequence contains the row sums.
- A110779 (program): Fractalization of e.
- A110800 (program): n-th digit after decimal point in decimal expansion of n/(n+1).
- A110801 (program): Numbers n such that 12n + 1 is prime.
- A110803 (program): n times the number of digits in the decimal expansion of n.
- A110804 (program): a(1) = 10, a(n) = a(n-1) times the number of digits in a(n-1).
- A110805 (program): Sum of digits of n times number of digits of n.
- A110807 (program): n times largest n-digit number.
- A110808 (program): Least factorial obtained as n(n-1)…(n-k).
- A110809 (program): a(1) = 3, a(2n) = a(2n-1)*(a(2n-1)+1)/2, a(2n+1) = a(2n)*(a(2n)-1)/2.
- A110810 (program): Binomial transform of A000796.
- A110812 (program): Fractalization of sqrt 2.
- A110813 (program): A triangle of pyramidal numbers.
- A110814 (program): Inverse of a triangle of pyramidal numbers.
- A110822 (program): G.f.: square root of weight enumerator of [16,5,8] Reed-Muller code RM(1,4).
- A110824 (program): G.f.: square root of weight enumerator of [32,6,16] Reed-Muller code RM(1,5).
- A110826 (program): G.f.: square root of weight enumerator of [64,7,32] Reed-Muller code RM(1,6).
- A110828 (program): G.f.: square root of weight enumerator of [128,8,64] Reed-Muller code RM(1,7).
- A110831 (program): a(n) = 3*n^2 + 27*n + 1.
- A110833 (program): a(n) = (prime(n)+1)^2.
- A110837 (program): Number of ways to fold a strip of n stamps taking account of order and direction of folds.
- A110847 (program): Weight enumerator of [32,31,2] Reed-Muller code RM(4,5).
- A110851 (program): Weight enumerator of [64,63,2] Reed-Muller code RM(5,6).
- A110854 (program): A155750(n)-A155067(n) = prime(2n+2)-prime(2n+1)-prime(2n)+prime(2n-1).
- A110858 (program): Triangle read by rows: number of order-preserving partial transformations (of an n-element chain) of width and waist both equal to r (width(alpha) = |Dom(alpha)| and waist(alpha) = max(Im(alpha)).
- A110862 (program): Highest minimal distance of odd formally self-dual binary codes of length 2n.
- A110867 (program): Highest minimal distance of Type I but not Type II additive Hermitian self-dual codes of length n over GF(4).
- A110870 (program): Highest minimal distance of Type II additive Hermitian self-dual codes of length n over GF(4).
- A110872 (program): Numbers n such that (n^2+7)/8 is prime.
- A110873 (program): Squares of the form 8p - 7, where p is prime.
- A110882 (program): a(n) is the least integer x such that x^n < 2 * (x-1)^n.
- A110883 (program): Sum of consecutive digits in the decimal expansion of Pi.
- A110892 (program): Sum of the squares of digits of n^2.
- A110893 (program): Numbers with a semiprime number of prime divisors (counted with multiplicity).
- A110895 (program): Number of integers between a(n) and a(n+1) equals the n-th prime.
- A110901 (program): Product_{k=1..n} (A013929(k)), the product of the first n positive integers that are each divisible by at least one square >= 4.
- A110903 (program): Difference between the factorial of n and the double factorial of n.
- A110906 (program): Expansion of (1 +34*x +121*x^2)/((1-x)*(x^2 -14*x +1)).
- A110907 (program): Number of points in the standard root system version of the D_3 (or f.c.c.) lattice having L_infinity norm n.
- A110914 (program): “Self-convolution mod 3” of central Delannoy numbers (see comment).
- A110916 (program): Number of squares between 10n and 10n+9 (inclusive).
- A110923 (program): Final two digits of prime(n), with leading zero omitted.
- A110934 (program): Difference between 3-almostprime(n) and 3-almostprime(n+2).
- A110935 (program): a(n) = if n mod 2 = 0 then 8*F(n)-n otherwise 8*F(n)-4, where F() = Fibonacci numbers A000045.
- A110936 (program): a(n) = denominator(Bernoulli(prime(n) - 1))/prime(n).
- A110947 (program): a(n) = permanent of an n X n matrix M of zeros and ones defined as follows: if we concatenate the rows of M to form a vector v of length n^2, v_i = 1 only if i = 1 or a multiple of 2.
- A110952 (program): Triangle read by rows: T(n,k) = number of permutations of [n] where the first increasing run has length k and the last increasing run has length n-k-1, 0<k<n-1.
- A110953 (program): Starting a priori with the fraction 1/1, the denominators of fractions built according to the rule: add top and bottom to get the new bottom, add top and 9 times the bottom to get the new top.
- A110954 (program): a(1) = 1; a(n) = nextprime(2.5*a(n-1)) for n > 1.
- A110959 (program): Numbers n such that 23*n^2 + 1 is prime.
- A110960 (program): Numbers n such that 23*n^2 + 4 is prime.
- A110961 (program): Numbers n such that 23*n^2 + 9 is prime.
- A110962 (program): Fractalization of A025480, zero-based version of Kimberling’s paraphrases sequence.
- A110963 (program): Fractalization of Kimberling’s paraphrases sequence beginning with 1.
- A110964 (program): Numbers k such that 23*k^2 + 16 is prime.
- A110965 (program): Numbers k such that 23*k^2 + 25 is prime.
- A110966 (program): Numbers k such that 23*k^2 + 36 is prime.
- A110967 (program): Numbers k such that 23*k^2 + 49 is prime.
- A110974 (program): Numbers n such that 23*n^2 - 1 is prime.
- A110976 (program): Sequence of numerators associated with the continued fraction based on the sequence d(n)= distance of n from closest prime ( A051699).
- A110977 (program): Sequence of denominators associated with the continued fraction based on the sequence d(n)= distance of n from closest prime ( A051699).
- A110994 (program): Numbers n such that 23*n^2 - 4 is prime.
- A110998 (program): Numbers n such that 23*n^2 - 9 is prime.
- A110999 (program): Numbers n such that 23*n^2 - 16 is prime.
- A111002 (program): a(n) = gcd(f(n), f(n+1)) where f(n) = n^4 + n^2 + 1.
- A111003 (program): Decimal expansion of Pi^2/8.
- A111006 (program): Another version of Fibonacci-Pascal triangle A037027.
- A111007 (program): Triangle T(n,m) which contains in row n the rounded ordinate value at abscissa m along the upper rim of the circle with diameter n centered at (n/2, 1).
- A111008 (program): a(n) = A000367(n)/A141590(n).
- A111017 (program): a(n) = (A102877(n+1) - A102877(n))/2.
- A111018 (program): Indices of Catalan numbers that are divisible by 3.
- A111019 (program): Indices of Catalan numbers that are == 1 mod 3 (cf. A000108).
- A111020 (program): Indices of Catalan numbers (A000108) that are == 2 mod 3.
- A111025 (program): Number of cubes between 10n and 10n+9 (inclusive).
- A111029 (program): Magic products of 3 X 3 multiplicative magic squares.
- A111033 (program): Sum of squares of first n digits of Pi.
- A111034 (program): Sum of squares of digits of e.
- A111040 (program): Numbers n such that 2*n^2 + 9 is prime.
- A111041 (program): Numbers n such that 2*n^2 + 25 is prime.
- A111043 (program): Partial sums of squares of digits of golden ratio phi (A001622).
- A111046 (program): Difference between squares of twin prime pairs.
- A111048 (program): a(n) = least i such that prime(n)/prime(n+1) < 1 - 1/i.
- A111049 (program): Triangle T(n,k), 0 <= k <= n, read by rows, given by [1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, …] DELTA [1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, …] where DELTA is the operator defined in A084938.
- A111051 (program): Numbers n such that 3*n^2 + 1 is prime.
- A111052 (program): Numbers n such that 3*n^2 + 4 is prime.
- A111053 (program): Number of permutations which avoid the patterns 1324 and (2143 with Bruhat restriction {2<->3}). Also the number of permutations whose graphs are acyclic.
- A111054 (program): Sum of squares of digits of decimal expansion of square root of 2.
- A111059 (program): Product{k=1 to n} (A005117(k)), the product of the first n squarefree positive integers.
- A111060 (program): a(n) = sum of primes dividing the n-th squarefree positive integer.
- A111061 (program): Begin with 1,2 In binary 1, 10. To get the sequence, left pad binary number with its precedent: 1,10, 110, 10110, 11010110, 1011011010110, etc. Note the number of bits of the n-th term is the (n-1)st Fibonacci number. Now convert back to decimal 1,2,6,22,214,5846, …
- A111062 (program): Triangle T(n, k) = binomial(n, k) * A000085(n-k), 0 <= k <= n.
- A111063 (program): a(0) = 1; a(n) = (n-1)*a(n-1) + n.
- A111066 (program): Numbers with digits 1 and 2 and at least one of each.
- A111068 (program): Numbers k such that 3*k^2 + 16 is prime.
- A111069 (program): Numbers k such that 3*k^2 + 25 is prime.
- A111071 (program): Difference between the product of two consecutive primes and the next prime.
- A111072 (program): Write the digit string 0123456789, repeated infinitely many times. Then, starting from the first “0” digit at the left end, move to the right by one digit (to the “1”), then two digits (to the “3”), then three digits (to the “6”), four digits (“0”), five digits (“5”), and so on. Partial sums of the digits thus reached are 0, 1, 4, 10, 10, 15, …
- A111074 (program): Let t(n) denote the triangular numbers (A000217). Sequence mixes t(n+2) and t(n).
- A111077 (program): Smallest squarefree integer > the n-th term of the Fibonacci sequence.
- A111080 (program): Sum of numbers under a triangle on a spiral staircase of width 10.
- A111082 (program): Numbers n such that 3*n^2 + 49 is prime.
- A111083 (program): Numbers k such that 3*k^2 + 64 is prime.
- A111087 (program): Neither primes nor semiprimes.
- A111089 (program): Largest prime factor of 2n.
- A111092 (program): Primes congruent to {1,69} mod 70.
- A111093 (program): Like sequence A111072 but moving right by the squares of the sequence of positive integers.
- A111094 (program): Numbers k such that 18*k + 1 is prime.
- A111096 (program): Partial sums of A137701.
- A111097 (program): Maximum likelihood estimate of the number of distinguishable marbles in an urn if repeated random sampling of one marble with replacement yields n different marbles before the first repeated marble.
- A111099 (program): Sum of even Fermat coefficients rounded to nearest integer.
- A111102 (program): Cumulative sum of squares of Kolakoski sequence (A000002).
- A111108 (program): a(n) = A001333(n) - (-2)^(n-1), n > 0.
- A111110 (program): Expansion of x*(x^4 - x^3 + 4x^2 - 3x + 1)/(1 - 5x + 9x^2 - 8x^3 + 2x^4 - x^5).
- A111113 (program): a(2^m) = 1, a(2^m+1) = -1 (m>0), otherwise a(n) = 0.
- A111114 (program): Integer part of prime(n)/pi(n).
- A111121 (program): a(n) = a(n-3)^3 + a(n-2)^2 + a(n-1); a(1) = -1, a(2) = 0, a(3) = 1.
- A111125 (program): Triangle read by rows: T(k,s) = ((2*k+1)/(2*s+1))*binomial(k+s,2*s), 0 <= s <= k.
- A111132 (program): a(n+1) = a(n) + (a(n) - a(n-1) + a(n) mod 10) mod 10 with a(0)=0 and a(1)=1.
- A111133 (program): Number of partitions of n into at least two distinct parts.
- A111135 (program): Product_{k=1..n} F(p(k)), where p(k) is the k-th prime and F(k) is the k-th Fibonacci number.
- A111136 (program): a(n) = Sum_{k=1..n} Fibonacci(prime(k)).
- A111138 (program): Let b(n) denote the number of nontriangular numbers less than or equal to n. Then a(n) = b(n-1) + a(b(n-1)), with a(1) = a(2) = 0, a(3) = 1.
- A111139 (program): a(n) = n!*Sum_{k=0..n} Fibonacci(k)/k!.
- A111140 (program): a(n) = (n!/(n+1))*Sum_{k=0..n} binomial(n+k-1,k)/k!.
- A111144 (program): a(n) = n*(n+13)*(n+14)/6.
- A111145 (program): Length of the Cunningham chain initiated by the n-th Sophie Germain prime.
- A111147 (program): Numbers n such that 5*n^2 + 1 is prime.
- A111148 (program): Numbers n such that 5*n^2 + 4 is prime.
- A111149 (program): Numbers n such that 5*n^2 + 9 is prime.
- A111150 (program): a(n) is the number of integers of the form (n+k)/|(n-k)| for k>0.
- A111153 (program): Sophie Germain semiprimes: semiprimes n such that 2n+1 is also a semiprime.
- A111160 (program): G.f.: C - Z; where C is the g.f. for the Catalan numbers (A000108) and Z is the g.f. for A055113 with offset 0.
- A111165 (program): Let qf(a,q) = Product(1-a*q^j,j=0..infinity); g.f. is qf(q,q^3)/qf(q^2,q^3).
- A111166 (program): Let p < q be consecutive primes; p is in the sequence if p/(q-p) is a record.
- A111168 (program): Semiprimes n such that 2*n - 1 is also a semiprime.
- A111170 (program): Semiprimes S such that 3*S + 1 is also a semiprime.
- A111171 (program): Semiprimes S such that 3*S - 1 is also a semiprime.
- A111174 (program): Numbers k such that 24*k + 1 is prime.
- A111175 (program): Numbers n such that 30*n + 1 is prime.
- A111177 (program): Number of base n numbers in which each digit appears at most once (all unnecessary 0’s deleted).
- A111179 (program): a(n) = Sum_{k=1..n} prime(k)!, where prime(k) is k-th prime.
- A111181 (program): Prime(n) - Pi(n).
- A111186 (program): Difference between the closest squares surrounding squarefree composite numbers.
- A111192 (program): Product of the n-th sexy prime pair.
- A111196 (program): a(n) = 2^(-n)*Sum_{k=0..n} binomial(2*n+1, 2*k+1)*A000364(n-k).
- A111199 (program): Numbers n such that 4k + 9 is prime.
- A111204 (program): Difference between the closest squares surrounding a squarefree composite number and n have a common divisor greater than 1.
- A111208 (program): Number of primes <= n-th triangular number.
- A111209 (program): Difference between the powers of two and the primes.
- A111213 (program): Difference between the closest squares surrounding prime p is prime.
- A111214 (program): Score for an n-letter word in the game of Boggle.
- A111215 (program): Numbers k such that 4k + 5 is prime.
- A111216 (program): a(n) = 31*a(n-1)-a(n-2).
- A111217 (program): d_7(n), tau_7(n), number of ordered factorizations of n as n = rstuvwx (7-factorizations).
- A111218 (program): d_8(n), tau_8(n), number of ordered factorizations of n as n = rstuvwxy (8-factorizations).
- A111219 (program): d_9(n), tau_9(n), number of ordered factorizations of n as n = rstuvwxyz (9-factorizations).
- A111220 (program): d_10(n), tau_10(n), number of ordered factorizations of n as n = rstuvwxyza (10-factorizations).
- A111221 (program): d_11(n), tau_11(n), number of ordered factorizations of n as n = rstuvwxyzab (11-factorizations).
- A111222 (program): Integers that can be expressed as the sum of 4 factorials.
- A111223 (program): Numbers n such that 5*n + 2 is prime.
- A111224 (program): Numbers n such that 5*n + 7 is prime.
- A111225 (program): Numbers n such that 5*n + 8 is prime.
- A111226 (program): Numbers n such that 5*n + 12 is prime.
- A111230 (program): Numbers n such that 5*n + 14 is prime.
- A111234 (program): a(1)=2; thereafter a(n) = (largest proper divisor of n) + (smallest prime divisor of n).
- A111235 (program): a(1)=a(2)=a(3)=a(4)=1. For n >= 5, a(n)= a(n-1)*a(n-2) + a(n-3)*a(n-4).
- A111236 (program): a(1)=a(2)=a(3)=a(4)=1. For n >= 5, a(n)= (a(n-1)+a(n-2)) * (a(n-3)+a(n-4)).
- A111249 (program): Numbers n such that 7*n + 8 is prime.
- A111250 (program): Numbers n such that 7*n + 10 is prime.
- A111251 (program): Numbers k such that 3*k^2 + 3*k + 1 is prime.
- A111254 (program): a(n) = Prime[n+2]+Prime[n]+1.
- A111255 (program): Primes in A111254.
- A111262 (program): a(n) = (1/n)*Sum_{k=1..n} F(4*k)*B(2*n-2*k)*binomial(2*n,2*k)), where F are Fibonacci numbers and B are Bernoulli numbers.
- A111277 (program): Number of permutations avoiding the patterns {2413,4213,2431,4231,4321}; also number of permutations avoiding the patterns {3142,3412,3421,4312,4321}; number of weak sorting class based on 2413 or 3142.
- A111279 (program): Number of permutations avoiding the patterns {3241,3421,4321}; number of weak sorting class based on 3241.
- A111281 (program): Number of permutations avoiding the patterns {2413,2431,4213,3412,3421,4231,4321,4312}; number of strong sorting class based on 2413.
- A111282 (program): Number of permutations avoiding the patterns {1432,2431,3412,3421,4132,4231,4312,4321}; number of strong sorting class based on 1432.
- A111283 (program): Number of permutations avoiding the patterns {4321, 45132, 45231, 35412, 53412, 45213, 43512, 45312, 456123, 451623, 356124}; number of strong sorting class based on 4321.
- A111284 (program): Number of permutations of [n] avoiding the patterns {2143, 2341, 2413, 2431, 3142, 3241, 3412, 3421, 4123, 4213, 4231, 4321, 4132, 4312}; number of strong sorting classes based on 2143.
- A111285 (program): Number of permutations avoiding the patterns {2431, 3421, 4231, 4321, 24513, 42513, 34512, 43512}; number of strong sorting class based on 2431.
- A111286 (program): Number of permutations avoiding the patterns {1342, 1432, 2341, 2431, 3142, 3241, 3412, 3421, 4132, 4231, 4312, 4321}; number of strong sorting class based on 1342.
- A111288 (program): a(1) = a(2) = a(3) = a(4) = 1. For n>= 5, a(n) = a(n-1)*a(n-3) + a(n-2)*a(n-4).
- A111289 (program): a(1) = a(2) = a(3) = a(4) = 1. For n>= 5, a(n) = a(n-1)*a(n-4) + a(n-2)*a(n-3).
- A111290 (program): a(1)=1, a(n) = n + (sum of distinct primes dividing a(n-1)).
- A111292 (program): Numbers n such that 6*n^2 + 6*n + 1 is prime.
- A111294 (program): Numbers n such that 23*n + 2 is prime.
- A111295 (program): Number of partitions of 3n+1.
- A111297 (program): First differences of A109975.
- A111305 (program): Composite numbers k such that a^(k-1) == 1 (mod k) only when a == 1 (mod k).
- A111306 (program): d_12(n), tau_12(n), number of ordered factorizations of n as n = rstuvwxyzabc (12-factorizations).
- A111312 (program): Numbers n such that 11*n + 2 is prime.
- A111314 (program): a(n) = a(n-1) + a(n-2) + 2 where a(0) = a(1) = 1.
- A111317 (program): Let f(a,q) = Product_{j>=0} (1 - a*q^j); g.f. is f(q^2,q^3) / f(q,q^3).
- A111318 (program): Numbers n such that 4 divides prime(1)+…+prime(n).
- A111329 (program): Number of partitions of T where T = (3n + 1) if n is even and T=(3n + 1)/2 if n is odd.
- A111330 (program): Let qf(a,q) = Product_{j >= 0} (1-a*q^j); g.f. is qf(q,q^4)/qf(q^3,q^4).
- A111333 (program): Number of odd numbers <= n-th prime.
- A111335 (program): Let qf(a,q) = Product(1-a*q^j,j=0..infinity); g.f. is qf(q^3,q^4)/qf(q,q^4).
- A111343 (program): G.f. A(x/(1-x)), where A = g.f. for A090351.
- A111350 (program): Squares n such that 2*n + 1 is a semiprime.
- A111351 (program): Semiprimes of the form 2*n + 1, where n is a square.
- A111352 (program): a(n+3) = a(n+2) + 3*a(n+1) + a(n).
- A111362 (program): Sequence defined by an recurrence.
- A111365 (program): a(n) = 5*a(n-1) + 3*a(n-2) where a(0) = a(1) = 1.
- A111367 (program): Numbers k such that 7*k + 5 is prime.
- A111368 (program): The number of maximal determinant {-1,1} matrices of order n.
- A111369 (program): Numbers k such that 13*k + 11 is prime.
- A111370 (program): Number of partitions of (6*n + 1).
- A111374 (program): Series expansion of the reciprocal of the Goellnitz-Gordon continued fraction.
- A111384 (program): a(n) = binomial(n,3) - binomial(floor(n/2),3) - binomial(ceiling(n/2),3).
- A111385 (program): a(n) = binomial(n,4) - binomial(floor(n/2),4) - binomial(ceiling(n/2),4).
- A111386 (program): a(1) = 1, a(2) = 3; for n >= 3, take a(n) to be the smallest odd number not occurring earlier such that a(n-1) divides the concatenation a(n-2)a(n).
- A111393 (program): Number of digits in n^3.
- A111394 (program): a(n) = product of first n integers not divisible by 3.
- A111395 (program): First digit of powers of 5 (n>=1).
- A111396 (program): a(n) = n*(n+7)*(n+8)/6.
- A111397 (program): Composite numbers (modulo 3).
- A111398 (program): Numbers which are the cube roots of the product of their proper divisors.
- A111405 (program): a(n) = f(f(n+1)) - f(f(n)), where f(0) = 0 and f(m) = bigomega(m) = A001222(m) for m > 0.
- A111406 (program): a(n) = f(f(n+1)) - f(f(n)), where f(m) = pi(m) = A000720(m), with f(0) = 0.
- A111407 (program): a(n) = f(f(n+1)) - f(f(n)), where f(0) = 0 and f(m) = tau(m) = A000005(m) for m > 0.
- A111408 (program): f(f(n+1))-f(f(n)), where f(0)=0, and for m>0, f(m) = sigma(m) = A000203(m).
- A111409 (program): a(n) = f(f(n+1)) - f(f(n)), where f(0)=0, and for m>0, f(m) = phi(m) = A000010(m).
- A111412 (program): f(f(n+1))-f(f(n)), where f(m) = wt(m) = A000120(m).
- A111417 (program): a(n) = A034869(n) - A008311(n).
- A111418 (program): Right-hand side of odd-numbered rows of Pascal’s triangle.
- A111424 (program): Sum_{i=1..n} (2i)!/i!.
- A111425 (program): a(n) = tribonacci(Fibonacci(n)).
- A111426 (program): Difference between largest and smallest prime factor of the n-th composite number.
- A111427 (program): Tribonacci(tetranacci(n)).
- A111428 (program): Tribonacci(pentanacci(n)).
- A111429 (program): Tribonacci(hexanacci(n)).
- A111430 (program): Tribonacci(heptanacci(n)).
- A111431 (program): a(n) = Fibonacci(tribonacci(n)).
- A111432 (program): Fibonacci(tetranacci(n)).
- A111433 (program): Fibonacci(pentanacci(n)).
- A111435 (program): a(n) = Fibonacci(hexanacci(n)).
- A111438 (program): Fibonacci(heptanacci(n)), restricted to nonzero heptanacci numbers.
- A111451 (program): Number of partitions of P where P=(5*n + 1) if n is even and P=((5*n + 1)/2) if n is odd.
- A111454 (program): a(n) = (n-4)^(n-3) - (n-3)^(n-4) + 1.
- A111455 (program): Numbers n such that 101*n + 97 is prime.
- A111457 (program): Number of semiprimes smaller than the n-th prime.
- A111458 (program): Numbers that cannot be represented as the sum of at most three Fibonacci numbers (with repetitions allowed).
- A111459 (program): Generalized Somos-4 sequence with a(n-2)^2 replaced by a(n-2)^5.
- A111466 (program): a(1) = 1, a(n+1) = a(n) - F(n+1), if F(n+1) <= a(n), else a(n+1) = a(n) + F(n+1). F(n) is the n-th Fibonacci number (A000045).
- A111484 (program): Triangular numbers all of whose digits are nonprimes.
- A111490 (program): Antidiagonal sums of the numerical array defined by M(n,k) = 1 + (k-1) mod n.
- A111491 (program): a(0) = 1; for n>0, a(n) = (2^n-1)*a(n-1)-(-1)^n.
- A111492 (program): Triangle read by rows: a(n,k) = (k-1)! * C(n,k).
- A111495 (program): Floor of 10^n/Li(10^n) - 1.
- A111500 (program): Number of squares in an n X n grid of squares with diagonals.
- A111501 (program): Numbers n such that n^3 - n^2 + 1 is prime.
- A111505 (program): Right half of Pascal’s triangle (A007318) with zeros.
- A111515 (program): Number of partitions of T where T=(7*n + 1) if n is even and T=((7*n + 1)/2) if n is odd.
- A111517 (program): Numbers n such that (7*n + 1)/2 is prime.
- A111526 (program): Number triangle T(n,k)=C((n+k)/2,k)(n+1)(1+(-1)^(n-k))/(2(k+1)); T(n,k)=(-1)^((n-k)/2)*A053120(n+1,k+1)/2^k; Riordan array ((1+x^2)/(1-x^2)^2,x/(1-x^2)).
- A111527 (program): Inverse of A111526. Row sums have general term C(n,floor(n/2))*(cos(Pi*n/2) + sin(Pi*n/2)).
- A111529 (program): Row 2 of table A111528.
- A111530 (program): Row 3 of table A111528.
- A111531 (program): Row 4 of table A111528.
- A111532 (program): Row 5 of table A111528.
- A111533 (program): Row 6 of table A111528.
- A111537 (program): Column 1 of triangle A111536.
- A111538 (program): Column 2 of triangle A111536; also equals column 0 of triangle A111541, which is the matrix log of triangle A111536.
- A111545 (program): Column 1 of triangle A111544.
- A111546 (program): Column 2 of triangle A111544.
- A111547 (program): Column 3 of triangle A111544; also found in column 0 of triangle A111549, which equals the matrix logarithm of A111544.
- A111554 (program): Column 1 of triangle A111553.
- A111555 (program): Column 2 of triangle A111553.
- A111556 (program): Column 3 of triangle A111553.
- A111566 (program): a(n) = ((1+sqrt(8))*(2+sqrt(2))^n + (1-sqrt(8))*(2-sqrt(2))^n)/2.
- A111567 (program): Binomial transform of A048654: generalized Pellian with second term equal to 4.
- A111568 (program): Triangle read by rows: row n contains n terms of the arithmetic progression having first term 1 and common difference 2[n^(n-1)-1]/(n-1).
- A111569 (program): a(n) = a(n-1) + a(n-3) + a(n-4) for n>3, a(0) = -1, a(1) = 1, a(2) = 2, a(3) = 1.
- A111570 (program): a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4.
- A111571 (program): a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4.
- A111572 (program): a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4.
- A111573 (program): a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4.
- A111574 (program): a(n) = a(n-1) + a(n-3) + a(n-4), n >= 4.
- A111575 (program): Powers of 3 repeated four times.
- A111587 (program): a(n) = 2*a(n-1) + 2*a(n-3) + a(n-4), a(0) = 1, a(1) = 4, a(2) = 9, a(3) = 20.
- A111589 (program): Triangle read by rows: number of idempotent order-preserving partial transformations (of an n-element totally ordered set) of width k (width(alpha) = |Dom(alpha)|
- A111595 (program): Triangle of coefficients of square of Hermite polynomials divided by 2^n with argument sqrt(x/2).
- A111596 (program): The matrix inverse of the unsigned Lah numbers A271703.
- A111597 (program): Lah numbers: a(n) = n!*binomial(n-1,6)/7!.
- A111598 (program): Lah numbers: a(n) = n!*binomial(n-1,7)/8!.
- A111599 (program): Lah numbers: a(n) = n!*binomial(n-1,8)/9!.
- A111600 (program): Lah numbers: a(n) = n!*binomial(n-1,9)/10!.
- A111601 (program): Exponential (binomial) convolution of A001818 (with interspersed zeros) and A000142 (factorials).
- A111602 (program): Third column (m=2) of unsigned triangle A111595.
- A111607 (program): Fourth column of A109626.
- A111636 (program): Triangle read by rows: T(n,k) (0<=k<=n) is the number of labeled graphs having k blue nodes and n-k green ones and only nodes of different colors can be joined by an edge.
- A111637 (program): Number of labeled graphs having n blue nodes and n green ones, where edges join only nodes of different colors.
- A111639 (program): Expansion of (3+8*x-3*x^2-2*x^3)/((x^2+4*x+1)*(x^2-2*x-1)).
- A111640 (program): Expansion of (-1+3*x+x^2-x^3)/((x^2+4*x+1)*(x^2-2*x-1)).
- A111641 (program): Expansion of -(1+x+3*x^2+x^3)/((x^2+4*x+1)*(x^2-2*x-1)).
- A111642 (program): Expansion of 2*(x-1)*(x+1)/((x^2+4*x+1)*(x^2-2*x-1)).
- A111643 (program): Expansion of 2*(x+1)^2/((x^2+4*x+1)*(x^2-2*x-1)).
- A111644 (program): Expansion of -(1+x^2)/((x^2+4*x+1)*(x^2-2*x-1)).
- A111645 (program): Expansion of (x+1)*(1-3*x)/((x^2+4*x+1)*(x^2-2*x-1)).
- A111647 (program): a(n) = A001541(n)*A001653(n)*A002315(n).
- A111648 (program): a(n) = A001541(n)^2 + A001653(n)^2 + A002315(n)^2.
- A111650 (program): 2n appears n times (n>0).
- A111651 (program): n appears 3n times.
- A111652 (program): 3n appears n times.
- A111653 (program): n-th composite number appears n times.
- A111654 (program): n appears n-th composite number times.
- A111655 (program): n-th composite number appears n-th composite number times.
- A111657 (program): n-th composite number appears n-th prime times.
- A111661 (program): Expansion of eta(q)^4 * eta(q^2) * eta(q^6)^5 / eta(q^3)^4 in powers of q.
- A111663 (program): Expansion of (-1+x^3+x^6+x^9)/((1-x)*(2*x-1)*(x^2+1)*(x^2+x+1)*(x^4-x^2+1)).
- A111665 (program): Expansion of (-1+x+2*x^2+5*x^4+3*x^3) / ((x-1)*(x+1)*(x^2-3*x+1)*(1+x^2)).
- A111666 (program): Expansion of (-2+3*x+3*x^2+4*x^3+3*x^4-5*x^5)/((x-1)*(x+1)*(1+x^2)*(x^2-3*x+1)).
- A111683 (program): n^k - n! where n^k > n! >= n^(k-1).
- A111684 (program): Least k such that the product of n consecutive integers beginning with k exceeds n^n.
- A111685 (program): n + n(n+1) + n(n+1)(n+2) + …, with n terms.
- A111686 (program): (n+1) + (n+1)(n+2) + …, with n terms.
- A111687 (program): Comprimorial(n): the product of the first n primes and the first n composite numbers.
- A111688 (program): Primes and composite numbers alternately in increasing order.
- A111690 (program): Least integer multiple of 1/n, truncated to n digits after decimal.
- A111693 (program): The number system may be represented by linearly stringing together all the square domains. The number of the domain is given by r. It is noted that this has the same value as the circuit number in the Ellerstein square spiral. One below each odd square is a zero-centered octagonal number, which is divisible by 8. The value of this is eight times a triangular number. It may be seen that there are r octads in each square domain. The sequence is the first prime number in the first octad of each square domain.
- A111694 (program): a(1) = 1+2-3 = 0, a(2) = 4+5+6-7 = 8, a(3) = 8+9+10+11-12 = 26, a(4) = 13+14+15+16+17-18 = 57, …
- A111695 (program): a(n) = C(n,a)*C(n,b)*C(n,c)… where n = abc… are the decimal digits of n.
- A111700 (program): Number of integers between p(n) and p(n+1) which are coprime to (p(n+1)-p(n)), where p(n) is the n-th prime.
- A111701 (program): Least integer obtained when n is divided by prime(1), then by prime(2), then by prime(3), …, stopping as soon as one of the primes does not divide it. In particular, a(2n-1) = 2n-1.
- A111706 (program): a(n) = concatenation of k times the k-th digit of n.
- A111707 (program): a(n) = Sum_{k = 1..ceiling(w/2)} d(k) * d(w+1-k), where (d(1), …, d(w)) is the decimal expansion of n.
- A111708 (program): a(n) = n concatenated with 9’s complement of n.
- A111710 (program): Consider the triangle shown below in which the n-th row contains the n smallest numbers greater than those in the previous row such that the arithmetic mean is an integer. Sequence contains the leading diagonal.
- A111711 (program): Leading column of triangle mentioned in A111710.
- A111712 (program): Arithmetic mean of the n-th row of triangle mentioned in A111710.
- A111713 (program): Number of reduced tree pairs of n-carets.
- A111715 (program): Sum of the squares of the first n squarefree numbers.
- A111721 (program): a(n) = a(n-1) + a(n-2) + 5 where a(0) = a(1) = 1.
- A111723 (program): Number of partitions of an n-set with an odd number of blocks of size 1.
- A111724 (program): Number of partitions of an n-set with an even number of blocks of size 1.
- A111728 (program): Decimal expansion of (11/4)^(1/3).
- A111732 (program): Sum of the squares of the first n nonsquarefree numbers (A013929).
- A111733 (program): a(n) = a(n-1) + a(n-2) + 7 where a(0) = a(1) = 1.
- A111734 (program): Expansion of (1-x)*(2*x^2+2*x+1) / ((x^2-x-1)*(x^2+x+1)).
- A111735 (program): Distance between k*(n-th prime) and next prime, k=3 case.
- A111736 (program): Distance between k*(n-th prime) and next prime, k=4 case.
- A111737 (program): Distance between k*(n-th prime) and next prime, k=5 case.
- A111738 (program): Distance between k*(n-th prime) and next prime, k=6 case.
- A111739 (program): Distance between k*(n-th prime) and next prime, k=7 case.
- A111740 (program): Distance between k*(n-th prime) and next prime, k=8 case.
- A111741 (program): Distance between k*(n-th prime) and next prime, k=9 case.
- A111742 (program): Distance between k*(n-th prime) and next prime, k=10 case.
- A111744 (program): a(2k-1) = k-th prime of form 1 mod 4, a(2k) = k-th prime of form 3 mod 4.
- A111745 (program): a(2k-1) = k-th prime congruent to 3 mod 4, a(2k) = k-th prime congruent to 1 mod 4.
- A111746 (program): Number of squares in the interior of the square with vertices (n,0), (0,n), (-n,0) and (0,-n) in a square (x,y)-grid.
- A111748 (program): a(n) = 1 if n-th composite number is squarefree, otherwise a(n) = 0.
- A111752 (program): Number of partitions of {1,..,n} into lists with an even number of lists of size 1, where a list means an ordered subset (cf. A000262).
- A111753 (program): Number of partitions of {1,..,n} into lists with an odd number of lists of size 1, where a list means an ordered subset, cf. A000262.
- A111755 (program): Excess of n over a greedy sum of distinct squares.
- A111766 (program): Numbers occurring in three Pythagorean triples of the form: odd: a, (a^2-1)/2, (a^2+1)/2 or even: a, a^2/4-1, a^2/4+1.
- A111774 (program): Numbers that can be written as a sum of at least three consecutive positive integers.
- A111775 (program): Number of ways n can be written as a sum of at least three consecutive integers.
- A111776 (program): Triangle read by rows: number of idempotent order-preserving partial transformations (of an n-element chain) of waist k (waist(alpha) = max(Im(alpha)).
- A111777 (program): Fourth column (m=3) of unsigned triangle A111595.
- A111778 (program): Fifth column (m=4) of unsigned triangle A111595.
- A111779 (program): Sixth column (m=5) of unsigned triangle A111595.
- A111780 (program): Seventh column (m=6) of unsigned triangle A111595.
- A111781 (program): Eighth column (m=7) of unsigned triangle A111595.
- A111782 (program): Ninth column (m=8) of unsigned triangle A111595.
- A111783 (program): Tenth column (m=9) of unsigned triangle A111595.
- A111784 (program): Eleventh column (m=10) of unsigned triangle A111595.
- A111802 (program): n^2-n-1 for n>3; a(1)=1; a(2)=2; a(3)=3.
- A111805 (program): Number triangle T(n,k)=binomial(2(n+k),4k).
- A111806 (program): Riordan array (1/(1+3x+2x^2),x/(1+3x+2x^2)).
- A111808 (program): Left half of trinomial triangle (A027907), triangle read by rows.
- A111850 (program): Number of numbers m <= n such that 0 equals the first digit after decimal point of square root of n in decimal representation.
- A111851 (program): Number of numbers m <= n such that 1 equals the first digit after decimal point of square root of n in decimal representation.
- A111852 (program): Number of numbers m <= n such that 2 equals the first digit after decimal point of square root of n in decimal representation.
- A111853 (program): Number of numbers m <= n such that 3 equals the first digit after decimal point of square root of n in decimal representation.
- A111854 (program): Number of numbers m <= n such that 4 equals the first digit after decimal point of square root of n in decimal representation.
- A111856 (program): Number of numbers m <= n such that 6 equals the first digit after decimal point of square root of n in decimal representation.
- A111857 (program): Number of numbers m <= n such that 7 equals the first digit after decimal point of square root of n in decimal representation.
- A111858 (program): Number of numbers m <= n such that 8 equals the first digit after decimal point of square root of n in decimal representation.
- A111859 (program): Number of numbers m <= n such that 9 equals the first digit after decimal point of square root of n in decimal representation.
- A111862 (program): Second digit after decimal point of square root of n in decimal representation.
- A111863 (program): Smallest prime factor of 6*n-1 that is congruent to 5 modulo 6.
- A111868 (program): The work performed by a function f:{1,…,n} -> {1,…,n} is defined to be work(f) = Sum_{i=1..n} |i - f(i)|; a(n) is equal to sum(work(f)) where the sum is over all functions f:{1,…,n}->{1,…,n}.
- A111873 (program): The work performed by a partial function f:{1,…,n}->{1,…,n} is defined to be work(f)=sum(|i-f(i)|,i in dom(f)); a(n) is equal to sum(work(f)) where the sum is over all partial functions f:{1,…,n}->{1,…,n}.
- A111874 (program): The work performed by a partial function f:{1,…,n}->{1,…,n} is defined to be work(f)=sum(|i-f(i)|,i in dom(f)); a(n) is equal to sum(work(f)) where the sum is over all injective partial functions f:{1,…,n}->{1,…,n}.
- A111876 (program): Denominator of Sum_{k = 0..n} 1/((k+1)*(2*k+1)).
- A111877 (program): Sequence related to f(n) = 1/1 + 1/3 + … + 1/(2n+1).
- A111878 (program): a(n) = denominator(digamma(n+7/2)/2 + log(2) + euler_gamma/2)/15; a(n)=denominator(f(n+2)/15) = A111877(n+1)/5.
- A111882 (program): Row sums of triangle A111595 (normalized rescaled squared Hermite polynomials).
- A111883 (program): Unsigned row sums of triangle A111595 (normalized rescaled squared Hermite polynomials).
- A111884 (program): E.g.f.: exp(x/(1+x)).
- A111889 (program): A repeated permutation of {0,…,8}.
- A111890 (program): Number of numbers m <= n such that 0 equals the second digit after decimal point of square root of n in decimal representation.
- A111893 (program): Number of numbers m <= n such that 3 equals the second digit after decimal point of square root of n in decimal representation.
- A111894 (program): Number of numbers m <= n such that 4 equals the second digit after decimal point of square root of n in decimal representation.
- A111903 (program): The work performed by a partial function f:{1,…,n}->{1,…,n} is defined to be work(f)=sum(|i-f(i)|,i in dom(f)); a(n) is equal to sum(work(f)) where the sum is over all order-preserving partial functions f:{1,…,n}->{1,…,n}.
- A111911 (program): a(n) = (4*n+1)!/( (2*n+1)! * ((n+1)!)^2 ).
- A111915 (program): Expansion of -x^2*(x-1)*(x^2-x+1)*(x+x^2+1)/(1-x^4+x^8).
- A111917 (program): The i-th term of the generalized Fibonacci sequence [0,k,k,2k,3k,…] is given by the formula F(i) = round( k/sqrt(5) * phi^i ) provided i >= s(k); a(n) = smallest value of k such that s(k) = n.
- A111918 (program): Numerator of x(n) = Sum_{k=1..n} ((odd part of k)/(k^3)).
- A111919 (program): Denominator of x(n) = Sum_{k=1..n} ((odd part of k)/(k^3)).
- A111920 (program): Numerator of x(n) = Sum_{k=1..n} ((odd part of k)/(k^4)).
- A111921 (program): Denominator of x(n) = Sum_{k=1..n} ((odd part of k)/(k^4)).
- A111922 (program): Numerator of x(n) = Sum_{k=1..n} ((odd part of k)/(k^5)).
- A111923 (program): Denominator of x(n) = Sum_{k=1..n} ((odd part of k)/(k^5)).
- A111924 (program): Triangle of Bessel numbers read by rows. Row n gives T(n,n), T(n,n-1), T(n,n-2), …, T(n,1) for n >= 1.
- A111926 (program): Expansion of x^4/((1-2*x)*(x^2-x+1)*(x-1)^2).
- A111927 (program): Expansion of x^3 / ((x-1)*(2*x-1)*(x^2-x+1)).
- A111928 (program): Numerator of f(n) := Product_{i=1..n} sigma(i)/i.
- A111929 (program): Numerator of x(n) = Sum_{k=1..n} ((odd part of k)/(k^2)).
- A111930 (program): Denominator of x(n) = Sum_{k=1..n} ((odd part of k)/(k^2)).
- A111932 (program): Expansion of q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.
- A111934 (program): Denominator of f(n) := Product_{i=1..n} sigma(i)/i.
- A111935 (program): Numerator of n-th term of the harmonic series after removal of all terms 1/m from Sum_{m=1..n} 1/m for which m contains a 9 in its decimal representation.
- A111936 (program): Denominator of n-th term of the harmonic series after removal of all terms 1/m from Sum_{m=1..n} 1/m for which m contains a 9 in its decimal representation.
- A111938 (program): a(n) = n times number of divisors of n of form 4m+1 - n times number of divisors of form 4m+3.
- A111939 (program): Number of primes < semiprime(n).
- A111940 (program): Triangle P, read by rows, that satisfies P^-1 = P(n+1,k+1) for n >= k >= 0, with P(k,k)=1 and P(k+1,1)=P(k+1,0) for k >= 0, where [P^-1] denotes the matrix inverse of P.
- A111942 (program): Column 0 of the matrix logarithm (A111941) of triangle A111940, which shifts columns left and up under matrix inverse; these terms are the result of multiplying the element in row n by n!.
- A111946 (program): Triangle read by rows: T(n,k) = gcd(Fibonacci(n), Fibonacci(k)), 1 <= k <= n.
- A111951 (program): Period 8: repeat [0,3,1,2,2,1,3,0].
- A111952 (program): a(n) = 3*n mod 7.
- A111954 (program): a(n) = A000129(n) + (-1)^n.
- A111955 (program): a(n) = A078343(n) + (-1)^n.
- A111956 (program): Triangle read by rows: T(n,k) = gcd(Lucas(n), Lucas(k)), 1 <= k <= n.
- A111957 (program): Triangle read by rows: T(n,k) = gcd(Fibonacci(n), Lucas(k)), 1 <= k <= n.
- A111958 (program): Lucas numbers (A000032) mod 8.
- A111959 (program): Renewal array for aerated central binomial coefficients.
- A111960 (program): Renewal array for central trinomial numbers A002426.
- A111961 (program): Expansion of 1/(sqrt(1-2x-3x^2)-x).
- A111962 (program): Expansion of 1/(sqrt(1-2x-3x^2)-x^2).
- A111963 (program): Inverse of renewal array for central trinomial numbers.
- A111965 (program): Riordan array (1/sqrt(1-6x+5x^2),x/(1-6x+5x^2)).
- A111966 (program): Expansion of 1/(sqrt(1-6x+5x^2)-x).
- A111968 (program): a(n) = Sum_{k=0..n} C(n,k)^2*C(n+k,k)^3, where C := binomial.
- A111972 (program): a(n) = Max(omega(k): 1<=k<=n), where omega(n) = A001221(n), the number of distinct prime factors of n.
- A111973 (program): Expansion of ((eta(q^2)eta(q^4))^6/(eta(q)eta(q^8))^4-1)/4 in powers of q.
- A111980 (program): Union of pairs of consecutive primes p, q with q-p = 4.
- A111981 (program): Numbers n such that 2n-1 and 2n+3 are consecutive primes.
- A111982 (program): Row sums of abs(A111967).
- A111983 (program): G.f.: A(x) = Sum_{n>=0} (2*n+1) * 8^n * x^(n*(n+1)/2).
- A111989 (program): G.f.: 1/(1-6*x+8*x^3).
- A111990 (program): Convolution of A111989 with itself.
- A111993 (program): Fifth convolution of Schroeder’s (second problem) numbers A001003(n), n>=0.
- A111994 (program): Sixth convolution of Schroeder’s (second problem) numbers A001003(n), n>=0.
- A111995 (program): Seventh convolution of Schroeder’s (second problem) numbers A001003(n), n >= 0.
- A111996 (program): Eighth convolution of Schroeder’s (second problem) numbers A001003(n), n>=0.
- A111997 (program): Ninth convolution of Schroeder’s (second problem) numbers A001003(n), n>=0.
- A111998 (program): Tenth convolution of Schroeder’s (second problem) numbers A001003(n), n>=0.
- A112000 (program): One half of third column (k=2) of triangle A111999.
- A112019 (program): a(n) = Sum_{k=0..n} binomial(n,k)*binomial(n+k,k)^2.
- A112028 (program): a(n) = Sum_{k=0..n} binomial(n+k,k)^3.
- A112029 (program): a(n) = Sum_{k=0..n} binomial(n+k, k)^2.
- A112030 (program): a(n) = (2 + (-1)^n) * (-1)^floor(n/2).
- A112031 (program): Numerator of 3/4 + 1/4 - 3/8 - 1/8 + 3/16 + 1/16 - 3/32 - 1/32 + 3/64 + ….
- A112032 (program): Denominator of 3/4 + 1/4 - 3/8 - 1/8 + 3/16 + 1/16 - 3/32 - 1/32 + 3/64 …
- A112033 (program): a(n) = 3 * 2^(floor(n/2) + 1 + (-1)^n).
- A112035 (program): a(n) = Sum_{k=0..n} k*C(n,k)^2*C(n+k,k)^3, where C := binomial.
- A112036 (program): a(n) = Sum_{k=0..n} k*C(n,k)^3*C(n+k,k), where C := binomial.
- A112039 (program): Let b(0)=1/2, b(n) = b(n-1) + Prime[n]/2; a(n)=b(2*n).
- A112040 (program): Terms in A112039 that are divisible by 3, divided by 3.
- A112044 (program): Let b(0)=1/2, b(n) = (b(n-1)+Prime[n])/2; sequence gives 2^(n+1)*b(n).
- A112045 (program): Positions of primes (A000040) among nonsquares A000037.
- A112051 (program): a(1)=1, a(n) = first index i (> a(n-1)), where A112046(i) gets a value distinct from any values A112046(1)..A112046(a(n-1)).
- A112052 (program): a(n) = 2*A112051(n)+1.
- A112062 (program): Positive integers i for which A112049(i) == 2.
- A112063 (program): Positive integers i for which A112049(i) == 3.
- A112072 (program): Odd numbers n for which 3 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.
- A112073 (program): Odd numbers n for which 5 is the smallest i (>= 1) with Jacobi symbol J(i,n) getting either a value 0 or -1.
- A112087 (program): a(n) = 4*(n^2 - n + 1).
- A112088 (program): Number of leaf nodes in a binary tree.
- A112091 (program): Number of idempotent order-preserving partial transformations (of an n-element chain).
- A112094 (program): Denominator of 3*Sum_{i=1..n} 1/(i^2*C(2*i,i)).
- A112097 (program): Numerator of Sum_{i=1..n} 1/C(2*i,i).
- A112098 (program): Denominator of Sum_{i=1..n} 1/C(2*i,i).
- A112099 (program): Numerator of Sum_{i=1..n} 1/(i*C(2*i,i)).
- A112100 (program): Denominator of Sum_{i=1..n} 1/(i*C(2*i,i)).
- A112101 (program): Number of interval orders of magnitude n having no duplicate holdings (NODH).
- A112102 (program): Numerator of Sum_{i=1..n} 1/(i^3*C(2*i,i)).
- A112103 (program): Denominator of Sum_{i=1..n} 1/(i^3*C(2*i,i)).
- A112128 (program): Expansion of phi(q^4) / phi(q) in powers of q where phi() is a Ramanujan theta function.
- A112132 (program): Period 4: repeat [1, 3, 1, 7].
- A112133 (program): First differences of A112063.
- A112141 (program): Product of the first n semiprimes.
- A112142 (program): McKay-Thompson series of class 8B for the Monster group.
- A112143 (program): McKay-Thompson series of class 8D for the Monster group.
- A112144 (program): McKay-Thompson series of class 8a for the Monster group.
- A112148 (program): McKay-Thompson series of class 12B for the Monster group.
- A112150 (program): McKay-Thompson series of class 16a for the Monster group.
- A112151 (program): McKay-Thompson series of class 16b for the Monster group.
- A112152 (program): McKay-Thompson series of class 16c for the Monster group.
- A112157 (program): McKay-Thompson series of class 18i for the Monster group.
- A112160 (program): McKay-Thompson series of class 24E for the Monster group.
- A112161 (program): McKay-Thompson series of class 24G for the Monster group.
- A112171 (program): McKay-Thompson series of class 32c for the Monster group.
- A112172 (program): McKay-Thompson series of class 32d for the Monster group.
- A112176 (program): McKay-Thompson series of class 36f for the Monster group.
- A112192 (program): Coefficients of replicable function number “48h”.
- A112205 (program): McKay-Thompson series of class 72a for the Monster group.
- A112227 (program): A scaled Hermite triangle.
- A112228 (program): Product of the first n (semiprimes - 1).
- A112231 (program): Repeat each prime in the sequence of natural numbers.
- A112232 (program): Repeat each composite number in the sequence of natural numbers.
- A112240 (program): Expansion of exp(x/(1-x-2x^2)).
- A112242 (program): E.g.f. exp( x*(1+x)/(1-x) ).
- A112243 (program): Expansion of exp(x*(1+x)/(1-2*x)).
- A112248 (program): a(n) = n mod floor(log_2(n)).
- A112249 (program): Numbers m such that m mod floor(log_2(m)) = 0.
- A112250 (program): Numbers m such that m mod floor(log_2(m)) > 0.
- A112251 (program): Numbers m such that m mod log_2(m) = 1.
- A112259 (program): Expansion of x*(1+8*x)/((1-8*x)*(1+11*x+64*x^2)).
- A112260 (program): Expansion of -x*(8*x^2-4*x+1) / ((2*x-1)*(4*x^2-x+1)).
- A112261 (program): a(n) = A112260(n+1) - A112260(n).
- A112270 (program): One third of the sum of the first n primes, when an integer.
- A112275 (program): Smallest number greater than n having at least as many divisors as n.
- A112278 (program): a(0) = 1; a(n) = prime(mod(a(n-1),100))+1.
- A112279 (program): a(1)=1; a(n)=prime(mod(a(n-1),100)).
- A112280 (program): Coefficients, read modulo 9, of the cube of q-series (q;q)_oo.
- A112282 (program): a(n) = (-1)^n*(2*n+1) (mod 9).
- A112292 (program): An invertible triangle of ratios of double factorials.
- A112293 (program): Row sums of number triangle A112292.
- A112294 (program): Diagonal sums of number triangle A112292.
- A112295 (program): Inverse of a double factorial related triangle.
- A112296 (program): Smret suoiverp eht fo mus fo esrever.
- A112298 (program): Expansion of (a(q) - 3*a(q^2) + 2*a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.
- A112299 (program): Expansion of x * (1 - x) * (1 - x^2) * (1 - x^3) / (1 - x^8) in powers of x.
- A112300 (program): Expansion of x * (1 - x)^2 * (1 - x^2) / (1 - x^6) in powers of x.
- A112301 (program): Expansion of (eta(q) * eta(q^16))^2 / (eta(q^2) * eta(q^8)) in powers of q.
- A112305 (program): Let T(n) = A000073(n+1), n >= 1; a(n) = smallest k such that n divides T(k).
- A112306 (program): a(n) = number of terms in s(n), where s(n) is defined in A096055.
- A112307 (program): Triangle read by rows: T(n,k) is number of Dyck paths of semilength n with height of second peak equal to k (n>=1; 0<=k<=n-1).
- A112308 (program): Sum of the heights of the second peaks in all Dyck paths of semilength n+2.
- A112310 (program): Number of terms in lazy Fibonacci representation of n.
- A112312 (program): Least index k such that the n-th prime divides the k-th tribonacci number.
- A112325 (program): Number of even semiprimes <= semiprime(n).
- A112326 (program): Triangle read by rows: T(n,k)=2^k*binomial(2n-k,n-k), 1<=k<=n.
- A112327 (program): Triangle read by rows: T(n,k)=k^3*2^k*binomial(2n-k,n-k)/(2n-k) (1<=k<=n).
- A112328 (program): a(n) = (n+1)*binomial(2n+2,n+1)-3*4^n+binomial(2n,n).
- A112329 (program): Number of divisors of n if n odd, number of divisors of n/4 if n divisible by 4, otherwise 0.
- A112332 (program): a(n) = Product_{k=0..n-1} k!*binomial(2k,k).
- A112333 (program): An invertible triangle of ratios of triple factorials.
- A112334 (program): Inverse of number triangle A112333.
- A112335 (program): Row sums of number triangle A112334.
- A112336 (program): A number triangle related to the central binomial coefficients.
- A112337 (program): a(1)=1, a(2) = 2. a(n) = a(n-2) + (largest prime dividing a(n-1)).
- A112341 (program): Number of primes between (prime(n)-1)^2 and prime(n)^2.
- A112342 (program): Number of primes between (n-th composite - 1)^2 and (n-th composite)^2.
- A112347 (program): Kronecker symbol (-1, n) except a(0) = 0.
- A112351 (program): Triangle read by rows, generated from (…, 5, 3, 1).
- A112352 (program): Triangular numbers that are the sum of two distinct positive triangular numbers.
- A112353 (program): Triangular numbers that are the sum of three distinct positive triangular numbers.
- A112355 (program): Triangular numbers that are the sum of three positive triangular numbers.
- A112358 (program): The following triangle is based on Pascal’s triangle. The r-th term of the n-th row is sum of C(n,r) successive integers so that the sum of all the terms of the row is (2^n)*(2^n+1)/2, the 2^n -th triangular number. Sequence contains the triangle read by rows.
- A112359 (program): Product of n-th row of A112358.
- A112367 (program): a(n) = A000217(n-k), where k is the largest triangular number less than n.
- A112368 (program): a(n) = Sum_{i=0..n} 2^i*i!.
- A112369 (program): -1 + Sum_{i=0..n} 2^i*i!.
- A112370 (program): Sum_{i=0..n} 3^i*i!.
- A112376 (program): Sum of base and exponent of prime powers.
- A112377 (program): A self-descriptive fractal sequence: if 1 is subtracted from every term and any zero terms are omitted, the original sequence is recovered (this process may be called “lower trimming”).
- A112378 (program): Adding 1 to every term produces the same sequence as omitting the 0’s.
- A112381 (program): Zero-free semiprimes.
- A112385 (program): a(n) = 6*binomial(4*n-1,n-1)/(4*n-1).
- A112387 (program): a(1)=1, a(2)=2, a(n)= 2^(n/2) if even and a(n-1)-a(n-2) if odd.
- A112391 (program): Primes p such that 23*p + 2 is also prime.
- A112392 (program): Squares of the form 3*k - 2 where k is a semiprime.
- A112393 (program): Semiprimes n such that 3*n - 2 is a square.
- A112399 (program): a(n) = Sum_{k=1..n, gcd(k,n)=1} mu(k), where mu(k) = A008683(k) (the Moebius function).
- A112403 (program): G.f.: (1-16*x+28*x^2+56*x^3-140*x^4+56*x^5+28*x^6-16*x^7+x^8)/(x^2-x+1)^8.
- A112413 (program): Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n and starting with exactly k UD’s, where U=(1,1), D=(1,-1) (0 <= k <= n).
- A112414 (program): 3n+7.
- A112415 (program): a(n) = C(1+n,1) * C(2+n,1) * C(4+n,2).
- A112416 (program): Next-to-most-significant binary digit of the n-th prime.
- A112421 (program): Number of 6 element subsets of {1,2,3,…,n} for which the sum-set has 12 elements.
- A112423 (program): Number of 6-element subsets of {1,2,3,…,n} which have a sum-set with 14 elements.
- A112440 (program): Next term is the sum of the last 10 digits in the sequence, beginning with a(10) = 9.
- A112447 (program): a(2*n) = A001045(n+2); a(2*n+1) = A001045(n+1).
- A112448 (program): a(n) = 1 if 2*n+1 is prime, otherwise a(n) = minimal residue of (-1)^binomial(n+2,2) mod (2n+1).
- A112455 (program): a(n) = -a(n-2) - a(n-3).
- A112456 (program): Least triangular number divisible by n-th prime.
- A112458 (program): Let b(n) = A112455(n). Then b(n)/n is an integer iff n is prime (at least for the first few values, as for the Perrin sequence). This sequence is the values of b(p)/p, where p is the n th prime.
- A112459 (program): Absolute value of coefficient of term [x^(n-3)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 3. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.
- A112460 (program): Absolute value of coefficient of term [x^(n-4)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 4. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.
- A112461 (program): Absolute value of coefficient of term [x^(n-5)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 5. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.
- A112462 (program): Absolute value of coefficient of term [x^(n-6)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 6. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.
- A112463 (program): Absolute value of coefficient of term [x^(n-7)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 7. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.
- A112464 (program): Absolute value of coefficient of term [x^(n-8)] in characteristic polynomial of maximum matrix A of size n X n, where n >= 8. Maximum matrix A(i,j) is MAX(i,j), where indices i and j run from 1 to n.
- A112465 (program): Riordan array (1/(1+x),x/(1-x)).
- A112466 (program): Riordan array ((1+2x)/(1+x), x/(1+x)).
- A112467 (program): Riordan array ((1-2x)/(1-x), x/(1-x)).
- A112468 (program): Riordan array (1/(1-x), x/(1+x)).
- A112469 (program): Partial sums of (-1)^n*F(n-1).
- A112475 (program): Riordan array (1/(1+x),x(1+x)/(1-x)).
- A112476 (program): Diagonal sums of Riordan array (1/(1+x),x(1+x)/(1-x)).
- A112477 (program): Riordan array ((1-x+sqrt(1+6x+x^2))/2, (sqrt(1+6x+x^2)-x-1)/2).
- A112478 (program): Expansion of (1 + x + sqrt(1 + 6*x + x^2))/2.
- A112484 (program): Array where n-th row contains the primes < n and coprime to n.
- A112488 (program): Third column of triangle A112486 used for e.g.f.s of |Stirling1| = |A008275| diagonals.
- A112489 (program): Fourth column of triangle A112486 used for e.g.f.s of |Stirling1|=|A008275| diagonals.
- A112494 (program): Sixth diagonal of the Stirling2 triangle A048993 and sixth column of triangle A008278.
- A112495 (program): Third column of triangle A112493 used for e.g.f.s of Stirling2 diagonals.
- A112498 (program): Third column of second-order Eulerian triangle A008517 divided by 2.
- A112502 (program): Third column of triangle A112500.
- A112503 (program): Fourth column of triangle A112500.
- A112504 (program): Fifth column of triangle A112500.
- A112508 (program): Counts the objects described in A047969 and A089246 when grouped by minimum part. (Row sums give A047970).
- A112509 (program): Maximum number of numbers represented by substrings of an n-bit number’s binary representation.
- A112517 (program): Riordan array (1,x(1+x)(1-x(1+x)).
- A112518 (program): Expansion of 1/(1-x+2x^3+x^4).
- A112520 (program): Expansion of 2/(3-sqrt(3-2*sqrt(1-4x))).
- A112521 (program): Sequence related to NOR bracketings.
- A112523 (program): Expansion of x*(1+3*x-4*x^2-5*x^3-4*x^6+4*x^5+3*x^4) / ((1+4*x^2)*(1+x^2)*(1-x^2+x^4)).
- A112524 (program): a(n) = a(n-1) + 2*n^2 with a(1) = 1.
- A112525 (program): Expansion of 1/(1 - 100*x^2 - 100*x^3).
- A112526 (program): Characteristic function for powerful numbers.
- A112532 (program): First differences of [0, A047970].
- A112539 (program): Half-baked Thue-Morse: at successive steps the sequence or its bit-inverted form is appended to itself.
- A112541 (program): a(n) = Sum_{k=0..n} (n-k)! * n^k.
- A112543 (program): Numerators of fractions n/k in array by antidiagonals.
- A112544 (program): Denominators of fractions n/k in array by antidiagonals.
- A112552 (program): A modified Chebyshev transform of the second kind.
- A112553 (program): Expansion of 1/( (1+x^2)*(1-x+x^2) ).
- A112554 (program): Riordan array (c(x^2)^2, x*c(x^2)), c(x) the g.f. of A000108.
- A112555 (program): Triangle T, read by rows, such that the m-th matrix power satisfies T^m = I + m*(T - I) and consequently the matrix logarithm satisfies log(T) = T - I, where I is the identity matrix.
- A112556 (program): Sums of squared terms in rows of triangle A112555.
- A112557 (program): Smallest number of stones in Tchoukaillon (or Mancala, or Kalahari) solitaire which make use of (2*n-1)-th hole for n>=1; a bisection of A002491.
- A112558 (program): Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 4 multiples of n-1, n-2, …, 1, for n>=1.
- A112559 (program): Numbers k such that both k and 4k + 1 are in A005098.
- A112560 (program): Sieve performed by successive iterations of steps where step m is: keep m terms, remove the next 2 and repeat; as m = 1,2,3,.. the remaining terms form this sequence.
- A112561 (program): Sieve performed by successive iterations of steps where step m is: keep m terms, remove the next 3 and repeat; as m = 1,2,3,.. the remaining terms form this sequence.
- A112562 (program): Sieve performed by successive iterations of steps where step m is: keep m terms, remove the next 4 and repeat; as m = 1,2,3,.. the remaining terms form this sequence.
- A112563 (program): Sieve performed by successive iterations of steps where step m is: keep m terms, remove the next 5 and repeat; as m = 1,2,3,.. the remaining terms form this sequence.
- A112565 (program): Main diagonal of square table A112564 of generalized Flavius Josephus sieves.
- A112566 (program): a(n) = (A112565(n) - 1)/n for n>=1.
- A112568 (program): Secondary diagonal of square table A112564 of generalized Flavius Josephus sieves.
- A112575 (program): Chebyshev transform of the second kind of the Pell numbers.
- A112576 (program): A Chebyshev-related transform of the Fibonacci numbers.
- A112577 (program): A Chebyshev-related transform of the Jacobsthal numbers.
- A112581 (program): Number of partitions of n into 5-smooth parts.
- A112591 (program): a(n) = prime(n) XOR prime(n + 1).
- A112594 (program): Natural function used for generating x^2 and sqrt(x) only using iteration and Q(x) (the characteristic function of sqrt).
- A112595 (program): Sequence of numerators of the continued fraction derived from the sequence of the number of distinct factors of a number (A001221, also called omega (n)).
- A112596 (program): Sequence of denominators of the continued fraction derived from the sequence of the numbers of distinct factors of a number (A001221, also called omega(n)).
- A112598 (program): a(1)=a(2)=1. For n >= 3, a(n) is smallest integer > a(n-1) such that gcd(a(n), a(n-1) + a(n-2)) > 1.
- A112603 (program): Number of representations of n as the sum of a square and a triangular number.
- A112604 (program): Number of representations of n as a sum of three times a square and two times a triangular number.
- A112605 (program): Number of representations of n as a sum of a square and six times a triangular number.
- A112606 (program): Number of representations of n as a sum of six times a square and a triangular number.
- A112607 (program): Number of representations of n as a sum of a triangular number and twelve times a triangular number.
- A112608 (program): Number of representations of n as a sum of a twice a square and three times a triangular number.
- A112609 (program): Number of representations of n as a sum of three times a triangular number and four times a triangular number.
- A112610 (program): Number of representations of n as a sum of two squares and two triangular numbers.
- A112618 (program): Let T(n) = A000073(n+1), n >= 1; a(n) = smallest k such that prime(n) divides T(k).
- A112620 (program): If b(n,1) = n; b(n,m) is number of terms among {b(n,1), b(n,2), …, b(n,m-1)} which are coprime to m, then a(n) = b(n,n).
- A112621 (program): If p^b(p,n) is the highest power of the prime p dividing n, then a(n) = sum_{p|n} b(p,n)^b(p,n).
- A112622 (program): If p^b(p,n) is the highest power of the prime p dividing n, then a(n) = product_{p|n} b(p,n)^b(p,n).
- A112623 (program): If p^b(p,n) is the highest power of the prime p dividing n, then a(n) = sum_{p|n} b(p,n)!.
- A112624 (program): If p^b(p,n) is the highest power of the prime p dividing n, then a(n) = Product_{p|n} b(p,n)!.
- A112626 (program): Triangle read by rows: T(n,k) = Sum_{j=0..n} binomial(n, k+j)*2^(n-k-j).
- A112627 (program): Decimal equivalent of number defined by last k bits of the infinite binary string …0011001100110011 (numbers with leading zeros omitted).
- A112632 (program): Excess of 3k - 1 primes over 3k + 1 primes, beginning with 2.
- A112638 (program): Power each digit individually according to its position and add the numbers.
- A112639 (program): a(n) = floor(r^n) where r is the smallest Pisot number (real root r=1.3247179… of x^3-x-1).
- A112651 (program): Numbers k such that k^2 == k (mod 11).
- A112652 (program): a(n) squared is congruent to a(n) (mod 12).
- A112653 (program): a(n) squared is congruent to a(n) (mod 13).
- A112654 (program): Numbers k such that k^3 == k (mod 11).
- A112655 (program): a(n) cubed is congruent to a(n) (mod 13).
- A112657 (program): A Motzkin transform of Jacobsthal numbers.
- A112658 (program): Dean’s Word: Omega 2,1: the trajectory of 0 -> 01, 1 -> 21, 2 -> 03, 3 -> 23.
- A112661 (program): Sum of digits of sum of previous 3 terms.
- A112667 (program): a(n+1) is the sum of the units digit of a(n) and the square of the tens digit of a(n).
- A112668 (program): Triangle read by rows: row n gives an n-term geometric progression with first term 1 such that the sum of the n terms is a multiple of n.
- A112677 (program): Sum of digits of the sum of the previous 4 terms.
- A112681 (program): Primes such that the sum of the predecessor and successor primes is divisible by 3.
- A112685 (program): a(n)=5a(n-2)+2a(n-3).
- A112689 (program): A modified Chebyshev transform of the Jacobsthal numbers.
- A112690 (program): Expansion of 1/(1+x^2-x^3-x^5).
- A112691 (program): a(n) = J(n+1) mod J(n), J(n)=A001045(n).
- A112693 (program): Row sums of array A112692.
- A112695 (program): Number of steps needed to reach 4,2,1 in Collatz’ 3*n+1 conjecture.
- A112696 (program): Partial sum of Catalan numbers A000108 multiplied by powers of 2.
- A112697 (program): Partial sum of Catalan numbers (A000108) multiplied by powers of 3.
- A112698 (program): Partial sum of Catalan numbers A000108 multiplied by powers of 4.
- A112699 (program): Partial sum of Catalan numbers A000108 multiplied by powers of 5.
- A112700 (program): Partial sum of Catalan numbers A000108 multiplied by powers of 6.
- A112701 (program): Partial sum of Catalan numbers A000108 multiplied by powers of 7.
- A112702 (program): Partial sum of Catalan numbers A000108 multiplied by powers of 8.
- A112703 (program): Partial sum of Catalan numbers A000108 multiplied by powers of 9.
- A112704 (program): Partial sum of Catalan numbers A000108 multiplied by powers of 10.
- A112705 (program): Triangle built from partial sums of Catalan numbers A000108 multiplied by powers.
- A112706 (program): Row sums of triangle A112705.
- A112707 (program): Triangle built from partial sums of Catalan numbers multiplied by powers of nonpositive numbers.
- A112708 (program): Row sums of triangle A112707 (partial sums of Catalan numbers multiplied by powers of negative numbers).
- A112709 (program): Unsigned row sums of triangle A112707 (partial sums of Catalan numbers multiplied by powers of negative numbers).
- A112710 (program): Partial sums of Catalan numbers A000108 multiplied by powers of -3.
- A112711 (program): Partial sums of Catalan numbers A000108 multiplied by powers of -4.
- A112712 (program): Expansion of x/(1 - x + 2*x^2 - 2*x^3 + 2*x^4 - x^5 + x^6).
- A112713 (program): Expansion of x/(1 - x + x^5 - x^6).
- A112714 (program): Numbers of the form k*2^m-1 with k<2^m and k odd.
- A112715 (program): Primes in A112714.
- A112739 (program): Array counting nodes in rooted trees of height n in which the root and internal nodes have valency k (and the leaf nodes have valency one).
- A112740 (program): Row sums of number triangle A112739.
- A112742 (program): a(n) = n^2*(n^2 - 1)/3.
- A112744 (program): Least k such that 6*k*prime(n)^2 -1 is prime, where prime(i)=i-th prime.
- A112745 (program): Least k such that 6*k*prime(n)^2 +1 is prime, where prime(i)=i-th prime.
- A112746 (program): Least k such that 6*k*prime(n)^2 - 1 and 6*k*prime(n)^2 + 1 are twin primes.
- A112751 (program): Number of numbers less than or equal to n of the form 3^i*5^j.
- A112757 (program): Greatest common divisors of consecutive 5-smooth numbers.
- A112758 (program): Number of distinct prime factors of n-th 5-smooth number.
- A112759 (program): Total number of prime factors of n-th 5-smooth number.
- A112760 (program): Exponent of 2 (value of i) in n-th number of the form 2^i*3^j*5^k.
- A112761 (program): Exponent of 3 (value of j) in n-th number of the form 2^i*3^j*5^k.
- A112762 (program): Exponent of 5 (value of k) in n-th number of the form 2^i*3^j*5^k.
- A112763 (program): Smallest prime factor of the n-th 5-smooth number.
- A112764 (program): Greatest prime factor of the n-th 5-smooth number.
- A112765 (program): Exponent of highest power of 5 dividing n. Or, 5-adic valuation of n.
- A112770 (program): Products of pairs of terms from A003627.
- A112771 (program): Semiprimes of the form 6n + 1.
- A112772 (program): Semiprimes of the form 6n+2.
- A112773 (program): 3 together with primes multiplied by 3.
- A112774 (program): Semiprimes of the form 6n+4.
- A112775 (program): Numbers n such that 6n+1 is semiprime.
- A112776 (program): Numbers n such that 6n+5 is semiprime.
- A112777 (program): Numbers n such that 2*n^2 + 1 is a semiprime.
- A112787 (program): Next term is the sum of previous term and the square of the sum of its decimal digits, with a(0) = 10.
- A112805 (program): Expansion of solution of functional equation.
- A112806 (program): Expansion of solution of functional equation.
- A112807 (program): Expansion of solution of functional equation.
- A112808 (program): Expansion of solution of functional equation.
- A112821 (program): Numbers n such that lcm(1,2,3,…,n)/19 equals the denominator of the n-th harmonic number H(n).
- A112823 (program): Greatest p less than or equal to n with p and q both prime, p+q = 2n.
- A112830 (program): Table of number of domino tilings of generalized Aztec pillows of type (1, …, 1, 3, 1, …, 1)_n.
- A112831 (program): Number of non-intersecting cycle systems in a particular directed graph.
- A112832 (program): Number of non-intersecting cycle systems in a particular directed graph.
- A112835 (program): Small-number statistic from the enumeration of domino tilings of a 3-pillow of order n.
- A112848 (program): Expansion of eta(q)*eta(q^2)*eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.
- A112849 (program): Number of congruence classes (epimorphisms/vertex partitionings induced by graph endomorphisms) of undirected cycles of even length: |C(C_{2*n})|.
- A112850 (program): Number of graph endomorphisms of undirected cycles of even length: |End(C_2n)|.
- A112851 (program): a(n) = (n-1)*n*(n+1)*(n+2)*(2*n+1)/40.
- A112857 (program): Triangle T(n,k) read by rows: number of Green’s R-classes in the semigroup of order-preserving partial transformations (of an n-element chain) consisting of elements of height k (height(alpha) = |Im(alpha)|).
- A112865 (program): a(n) = (-1)^(n + floor(n/4) + floor(n/4^2) + …).
- A112867 (program): Greater of two ternary (base 3) numbers (each using only 0’s and 1’s, the latter’s positions never coinciding) such that the decimal representation of their difference is n.
- A112873 (program): Partial sums of A032378.
- A112883 (program): A skew Jacobsthal-Pascal matrix.
- A112884 (program): Number of bits required to represent binomial(2^n, 2^(n-1)).
- A112885 (program): Primes of the form k + prime(k-1).
- A112886 (program): Positive integers that have no triangular divisors > 1.
- A112887 (program): Semiprime(n) - composite(n).
- A112899 (program): A skew Pell-Pascal triangle.
- A112925 (program): Largest squarefree integer < the n-th prime.
- A112926 (program): Smallest squarefree integer > the n-th prime.
- A112929 (program): Number of squarefree integers less than the n-th prime.
- A112930 (program): a(n) = order of n-th term of A112926 among squarefree integers.
- A112934 (program): a(0) = 1; a(n+1) = Sum_{k=0..n} a(k)*A001147(n-k), where A001147 = double factorial numbers.
- A112935 (program): Logarithmic derivative of A112934 such that a(n)=(1/2)*A112934(n+1) for n>0, where A112934 equals the INVERT transform of double factorials A001147.
- A112936 (program): INVERT transform (with offset) of triple factorials (A008544), where g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^3]/A(x)^3.
- A112937 (program): Logarithmic derivative of A112936 such that a(n)=(1/3)*A112936(n+1) for n>0, where A112936 equals the INVERT transform (with offset) of triple factorials A008544.
- A112938 (program): INVERT transform (with offset) of quadruple factorials (A008545), where g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^4]/A(x)^4.
- A112939 (program): Logarithmic derivative of A112938 such that a(n)=(1/4)*A112938(n+1) for n>0, where A112938 equals the INVERT transform (with offset) of quadruple factorials A008545.
- A112940 (program): INVERT transform (with offset) of quintuple factorials (A008546), where g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^5]/A(x)^5.
- A112941 (program): Logarithmic derivative of A112940 such that a(n)=(1/5)*A112940(n+1) for n>0, where A112940 equals the INVERT transform (with offset) of quintuple factorials A008546.
- A112942 (program): INVERT transform (with offset) of sextuple factorials (A008543), where g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^6]/A(x)^6.
- A112943 (program): Logarithmic derivative of A112942 such that a(n)=(1/6)*A112942(n+1) for n>0, where A112942 equals the INVERT transform (with offset) of sextuple factorials A008543.
- A112952 (program): Smaller of two ternary (base 3) numbers (each using only 0’s and 1’s, the latter’s positions never coinciding) such that the decimal representation of their difference is n.
- A112953 (program): a(1) = 0; a(n) = pi(n)^pi(n) for n>1, where pi is the prime counting function (A000720).
- A112957 (program): a(1) = a(2) = a(3) = 1; for n > 1, a(n+3) = a(n)^2 + a(n+1)^2 + a(n+2)^2.
- A112958 (program): a(1) = a(2) = a(3) = a(4) = 1; for n>1: a(n+4) = a(n)^2 + a(n+1)^2 + a(n+2)^2 + a(n+3)^2.
- A112963 (program): Sum(mu(i)*tau(j): i+j=n), with mu=A008683 and tau=A000005.
- A112964 (program): Sum(mu(i)*sigma(j): i+j=n), with mu=A008683 and sigma=A000203.
- A112967 (program): Sum(Omega(i)*Omega(j): i+j=n), with Omega=A001222.
- A112970 (program): A generalized Stern sequence.
- A112971 (program): Row sums of the matrix ((1,xc(x))^2 mod 2), where c(x) is the g.f. of A000108.
- A112973 (program): Riordan array (1/(1-x-x^2), x(1+x)/(1-x-x^2)^2).
- A112976 (program): Position of n-th prime in A112975.
- A112983 (program): 2^(n+1) mod n.
- A112984 (program): Numbers k such that 2^k mod k-1 is odd.
- A112985 (program): 2^(2^n mod n-1).
- A112986 (program): Crossing number of K_{4,n} on the real projective plane.
- A112987 (program): a(n) = 2^(2^n mod n) for n > 0; a(0) = 2.
- A112988 (program): Position of n-th prime in A089088.
- A112991 (program): a(n)=ceiling(2^(mod(2^n,n)+1)/3)-ceiling(2^mod(2^n,n)/3).
- A112997 (program): Sum of first n primes minus sum of their indices.
- A113009 (program): {Sum of the digits of n} raised to the power {number of digits of n}.
- A113010 (program): {Number of digits of n} raised to the power of {the sum of the digits of n}.
- A113011 (program): Decimal expansion of 1/(e^(1/2)-1).
- A113012 (program): Numerators of convergents to 1 + 2/(3 + 4/(5 + 6/(7 + …))).
- A113013 (program): Denominators of convergents to 1 + 2/(3 + 4/(5 + 6/(7 + …))).
- A113014 (program): Decimal expansion of value of the continued fraction 1/(2+3/(4+5/(6+7/….
- A113016 (program): Primes that remain primes when their decimal representation is interpreted duodecimally.
- A113020 (program): Number triangle whose row sums are the Fibonacci numbers.
- A113021 (program): Expansion of x^2/(1 - 2*x + 2*x^2 - x^3 - x^4).
- A113022 (program): a(n) = size of union of 2^k (mod 10^n), 0 < k <= 5^n.
- A113023 (program): Number of terms in A095810 which have n digits.
- A113025 (program): Triangle of integer coefficients of polynomials P(n,x) of degree n, and falling powers of x, arising in diagonal Padé approximation of exp(x).
- A113029 (program): a(1) = 2, a(2) = 3; for n > 2, a(n) = least prime equal to the sum of two or more previous terms.
- A113032 (program): a(n) = Sum_{k=0..floor(n/8)} binomial(n-5*k, 3*k).
- A113045 (program): Number triangle binomial(n,floor((n-k)/2)) mod 3.
- A113046 (program): Diagonal sums of number triangle binomial(n, floor((n-k)/2)) mod 3.
- A113047 (program): a(n) = C(3n,n)/(2n+1) mod 3.
- A113048 (program): Binomial(4n,n)/(3n+1) mod 4.
- A113049 (program): Triangle of sums of Jacobsthal numbers related to binomial(4n,n)/(3n+1) mod 4.
- A113050 (program): a(1) = a(2) = 1; for n>2, a(n+1) = a(n) + a(n-1) iff n is prime, otherwise a(n+1) = a(n) + 1.
- A113051 (program): a(1) = a(2) = 1; for n>2, a(n+1) = a(n) + a(n-1) iff a(n) is prime, otherwise a(n+1) = a(n) + 1.
- A113052 (program): Binomial(5n,n)/(4n+1) mod 5.
- A113059 (program): a(n) = n!*Sum_{k=0..n} A000296(k)/k!, n=0,1,… .
- A113060 (program): a(n) = n!*Sum_{k=0..n} bell(k+1)/k!, n=0,1…, where bell(n) are the Bell numbers, cf. A000110.
- A113061 (program): Sum of cube divisors of n.
- A113062 (program): Expansion of theta series of hexagonal net with respect to a node.
- A113063 (program): Associated with theta series of hexagonal net with respect to a node.
- A113066 (program): Expansion of (1 + x)^2/((1 + x + x^2)*(1 + 3*x + x^2)).
- A113067 (program): Expansion of -x/((x^2+x+1)*(x^2+3*x+1)); Invert transform gives signed version of Tetrahedral numbers A000292.
- A113070 (program): Expansion of ((1+x)/(1-2x))^2.
- A113071 (program): Expansion of ((1+x)/(1-3*x))^2.
- A113072 (program): Tridiagonal matrix associated with coordination sequences.
- A113115 (program): Primes p such that 17*p + 2 is also prime.
- A113117 (program): a(1) = 2; for n>1, a(n) is the smallest integer > a(n-1) such that all primes <= a(n-1) divide at least one integer k for a(n-1) < k <= a(n).
- A113119 (program): Total number of digits in all n-digit nonnegative integers.
- A113125 (program): A simple tridiagonal matrix.
- A113126 (program): A simple 4-diagonal matrix.
- A113127 (program): Expansion of (1 + x + x^2 + x^3)/(1-x)^2.
- A113128 (program): A simple 4-diagonal matrix based on (1+x)^3.
- A113139 (program): Number triangle, equal to half of Delannoy square array A008288.
- A113141 (program): Inverse of a Delannoy related triangle.
- A113142 (program): Expansion of x(1-3x+x^2+x^3)/(1+x)^2.
- A113144 (program): Row 3 of table A113143; equal to INVERT of triple (or 3-fold) factorials shifted one place right.
- A113145 (program): Row 4 of table A113143; equal to INVERT of quartic (or 4-fold) factorials shifted one place right.
- A113146 (program): Row 5 of table A113143; equal to INVERT of quintic (or 5-fold) factorials shifted one place right.
- A113147 (program): Row 6 of table A113143; equal to INVERT of 6-fold factorials shifted one place right.
- A113148 (program): Row 7 of table A113143; equal to INVERT of 7-fold factorials shifted one place right.
- A113149 (program): Row 8 of table A113143; equal to INVERT of 8-fold factorials shifted one place right.
- A113151 (program): Primes p such that 19*p + 2 is also prime.
- A113161 (program): a(1) = 1, a(n+1) = largest prime <= a(n)+n.
- A113166 (program): Total number of white pearls remaining in the chest - see Comments.
- A113167 (program): Triangle read by rows; n-th row begins with n and contains n primes greater than n and not already used.
- A113168 (program): Sum of digits of first n palindromes.
- A113169 (program): Primes p such that 13*p + 2 is also prime.
- A113170 (program): Ascending descending base exponent transform of odd numbers A005408.
- A113175 (program): Replace each prime p in prime-factorization of n with p-th Fibonacci number.
- A113176 (program): Product_{p|n} F(p), where F(p) is the p-th Fibonacci number and where the product is over the distinct prime divisors of n.
- A113177 (program): If, for p prime, p^(m_{n,p}) is the highest power of p dividing n with m>=0, then a(n) = Sum_{p prime} F(p)*(m_{n,p}), where F(p) = p-th Fibonacci number.
- A113178 (program): Sum_{p|n} F(p), where F(p) is the p-th Fibonacci number and where the sum is over the distinct prime divisors of n.
- A113179 (program): Expansion of 1/sqrt((1-2x)^2-8x^3).
- A113180 (program): Expansion of 1/sqrt((1-2*x)^2-8*x^4).
- A113183 (program): Number of unrooted two-face maps in the plane (considered up to orientation-preserving homeomorphism) with the faces of equal degree n: planar maps with a distinguished outside face.
- A113184 (program): Absolute difference between sum of odd divisors of n and sum of even divisors of n.
- A113186 (program): Expansion of (25phi(q)phi^3(q^5)-phi^5(q)/phi(q^5)-24)/40 in powers of q where phi(q) is a Ramanujan theta function.
- A113187 (program): Inverse of twin-prime related triangle A111125.
- A113214 (program): Riordan array (1+2x,x(1+x)).
- A113215 (program): Repeat A006218(n) 2n+1 times.
- A113216 (program): Triangle of polynomials P(n,x) of degree n related to Pi (see comment) and derived from Padé approximation to exp(x).
- A113217 (program): Parity of decimal digital root of n.
- A113224 (program): a(2n) = A002315(n), a(2n+1) = A082639(n+1).
- A113225 (program): a(2n) = A011900(n), a(2n+1) = A001109(n+1).
- A113231 (program): Ascending descending base exponent transform of triangular numbers (A000217).
- A113235 (program): Number of partitions of {1,..,n} into any number of lists of size not equal to 2, where a list means an ordered subset, cf. A000262.
- A113236 (program): Number of partitions of {1,..,n} into any number of lists of size not equal to 3, where a list means an ordered subset, cf. A000262.
- A113237 (program): E.g.f.: exp(x*(1 - x^3 + x^4)/(1-x)).
- A113240 (program): Expansion of (1/(1-x))*sum(k>=2,x^k/(1-2x^k)).
- A113241 (program): Sum{k=1..n, tau(2k)-1}.
- A113242 (program): Numbers of the form 3^i +/- 3^j.
- A113245 (program): a(n) = floor(binomial(2n,2k)/binomial(n,k)).
- A113246 (program): a(2^n+a) = a(2^(n-1)+a) [if 0 <= a < 2^(n-1)], 3^n-a(2^n-a) [if 2^(n-1) <= a < 2^n].
- A113247 (program): Number of permutations pi in S_n such that maj pi and maj pi^(-1) have the same parity where maj is the major index. Equivalently, the number of pi such that maj pi and inv pi have the same parity where inv is the inversion number.
- A113248 (program): Number of permutations pi in S_n such that maj pi and maj pi^(-1) have opposite parity where maj is the major index. Equivalently, the number of pi such that maj pi and inv pi have opposite parity where inv is the inversion number.
- A113259 (program): Expansion of psi(x)^5 / psi(x^5) - 25*x^2 * psi(x) * psi(x^5)^3 in powers of x where psi() is a Ramanujan theta function.
- A113260 (program): Expansion of (-1 + psi(q)^5/psi(q^5) - 25q^2 psi(q)*psi(q^5)^3)/5 in powers of q where psi(q) is a Ramanujan theta function.
- A113261 (program): Expansion of (9*phi(q)*phi(q^3)^5 - phi(q)^5*phi(q^3))/8 in powers of q where phi(q) is a Ramanujan theta function.
- A113262 (program): One quarter of the number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.
- A113264 (program): Partial sums of Catalan numbers A000108 multiplied by powers of -5.
- A113265 (program): Partial sums of Catalan numbers A000108 multiplied by powers of -6.
- A113266 (program): Partial sums of Catalan numbers A000108 multiplied by powers of -7.
- A113267 (program): Partial sums of Catalan numbers A000108 multiplied by powers of -8.
- A113268 (program): Partial sums of Catalan numbers A000108 multiplied by powers of -9.
- A113269 (program): Partial sums of Catalan numbers A000108 multiplied by powers of -10.
- A113271 (program): Ascending descending base exponent transform of 2^n.
- A113276 (program): Decimal expansion of de Bruijn’s constant.
- A113277 (program): Expansion of q^(-1/3) * eta(q^2)^5 / eta(q)^2 in powers of q.
- A113278 (program): Triangle T, read by rows, such that the matrix square, T^2, forms a simple 2-diagonal triangle where T^2 = 1 and T^2 = 2*(n+1) for n>=0.
- A113280 (program): A symmetrical triangle of coefficients: t(n,m)=(n - m)*(n - m + 2)*m*(m + 2) + 1.
- A113281 (program): Self-convolution equals A113224.
- A113282 (program): Logarithmic derivative of the g.f. of A113281.
- A113283 (program): Even bisection of A113281: a(n) = A113281(2*n).
- A113284 (program): Odd bisection of A113281: a(n) = A113281(2*n+1).
- A113291 (program): a(n) = A113290(n,1)/(n+1) for n>=0, where A113290 is the matrix log of triangle A113287.
- A113292 (program): Column 0 of triangle A113290, which is the matrix log of A113287.
- A113296 (program): Cumulative product of double factorial A006882.
- A113300 (program): Sum of even-indexed terms of tribonacci numbers.
- A113301 (program): Sum of odd-indexed terms of tribonacci numbers.
- A113306 (program): Expansion of q * f(-q, -q^11) / f(-q^5, -q^7) in powers of q where f(, ) is Ramanujan’s general theta function.
- A113310 (program): Riordan array ((1+x)/(1-x),x/(1+x)).
- A113311 (program): Expansion of (1+x)^2/(1-x).
- A113312 (program): Expansion of (1+x)^2/(1-2x^2+x^3).
- A113313 (program): Riordan array (1-2x,x/(1-x)).
- A113321 (program): Lexicographically earliest permutation of the natural numbers such that all positive differences between succeeding terms occur exactly once.
- A113322 (program): First differences of A113321.
- A113323 (program): A113321(A113321(n)).
- A113324 (program): Inverse integer permutation of A113321.
- A113325 (program): A113324(A113324(n)).
- A113327 (program): a(n) = Sum_{k=0..n} 2^k*A111146(n,k).
- A113328 (program): a(n) = Sum_{k=0..n} 3^k*A111146(n,k).
- A113329 (program): a(n) = Sum_{k=0..n} 4^k*A111146(n,k).
- A113335 (program): a(n) = 3^5 * binomial(n+4, 5).
- A113337 (program): Number of noncrossing partitions of [n] with all blocks of odd size and 1 and n in the same block.
- A113338 (program): Positive integers of the form (18*m^2+1)/11.
- A113396 (program): Prime(n+1)^2-prime(n)^2 (mod prime(n+1)).
- A113397 (program): What are the values of k in the term Prime(n+1)^2-Prime(n)^2 = a+k*(Prime(n+1)) if “a” is element of {0,1,…,Prime(n+1)-1}.
- A113402 (program): Algebraic degree of cos(Pi/n) for constructible n-gons (A003401).
- A113405 (program): Expansion of x^3/(1 - 2*x + x^3 - 2*x^4) = x^3/( (1-2*x)*(1+x)*(1-x+x^2) ).
- A113406 (program): Half the number of integer solutions to x^2 + 4 * y^2 = n.
- A113407 (program): Expansion of psi(x) * phi(x^2) in powers of x where psi(), phi() are Ramanujan theta functions.
- A113409 (program): A transform of the central binomial coefficients A001405.
- A113411 (program): Excess of number of divisors of 2n+1 of form 8k+1, 8k+3 over those of form 8k+5, 8k+7.
- A113413 (program): A Riordan array of coordination sequences.
- A113414 (program): Expansion of Sum_{k>0} x^k/(1-(-x^2)^k).
- A113415 (program): Expansion of Sum_{k>0} x^k/(1-x^(2k))^2.
- A113416 (program): Expansion of eta(q^2)^7*eta(q^4)/(eta(q)*eta(q^8)^2) in powers of q.
- A113417 (program): Expansion of phi(x) * phi(-x)^2 * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.
- A113418 (program): Expansion of (eta(q^2)^7*eta(q^4)/(eta(q)*eta(q^8))^2-1)/2 in powers of q.
- A113419 (program): Expansion of phi(x)^2 * phi(-x) * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.
- A113422 (program): a(n) = floor((5*n^2+1)/3).
- A113424 (program): a(n) = (6n)!/((3n)!(2n)!n!).
- A113428 (program): Expansion of f(-x^2, -x^3) in powers of x where f(, ) is Ramanujan’s general theta function.
- A113429 (program): Expansion of f(-x, -x^4) in powers of x where f(, ) is Ramanujan’s general theta function.
- A113435 (program): a(n) = a(n-1) + Sum_{k=0..n/3} a(n-3k) with a(0)=1.
- A113436 (program): First row of A113435.
- A113437 (program): Second row of A113435.
- A113438 (program): Third row of A113435.
- A113439 (program): a(n) = a(n-1) + Sum_{k=1..floor(n/4)} a(n-4k), with a(0)=1.
- A113440 (program): First row of A113439.
- A113441 (program): Second row of A113439.
- A113442 (program): Third row of A113439.
- A113443 (program): Fourth row of A113439.
- A113444 (program): a(n) = a(n-1) + Sum_{0<k<=n/5} a(n-5k) with a(0)=1.
- A113446 (program): Expansion of (phi(q)^2 - phi(q^3)^2) / 4 in powers of q where phi() is a Ramanujan theta function.
- A113447 (program): Expansion of i * theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.
- A113448 (program): Expansion of (eta(q^2)^2 * eta(q^9) * eta(q^18)) / (eta(q) * eta(q^6)) in powers of q.
- A113449 (program): Sum of the square root of n-th square triangular number and n-th Pell (or lambda) number (A000129).
- A113450 (program): Difference between the square root of n-th square triangular number and n-th lambda number given by the recurrence f(n) = 2f(n-1) + f(n-2), f(1) = 1, f(2)= 2.
- A113452 (program): a(n) is the n-th smallest permanental minor of any H_m (m >= n), where H_m is the square matrix of order m with 1’s on or below the super diagonal and 0’s elsewhere.
- A113453 (program): Triangle giving maximal permanent P(n,k) of an n X n lower Hessenberg (0,1)-matrix with exactly k 1’s for 2 <= n <= k <= 2n, read by rows.
- A113459 (program): Least number that begins an arithmetic progression of n numbers with the same prime signature.
- A113473 (program): n repeated 2^(n-1) times.
- A113474 (program): a(n) = a(floor(n/2)) + floor(n/2) with a(1) = 1.
- A113476 (program): Decimal expansion of (log(2) + Pi/sqrt(3))/3.
- A113479 (program): Starting with the fraction 4/1 as the first term, a(n) is the numerator of the reduced fraction of the n-th term according to the rule: if n is even, multiply the previous term by n/(n+1); otherwise multiply the previous term by (n+1)/n.
- A113486 (program): a(n) = A113166(n) - Fibonacci(n-1), where Fibonacci(n) = A000045(n).
- A113487 (program): Numbers k such that 17*k + 2 is prime.
- A113488 (program): Numbers k such that 19*k + 2 is prime.
- A113497 (program): Ascending descending base exponent transform of sequence A000034(n) = 1 + n mod 2.
- A113502 (program): A number n is included if at least one of its divisors > 1 is a triangular number (i.e., is of the form m(m+1)/2, m >= 2).
- A113510 (program): Numbers k such that 29*k + 2 is prime.
- A113519 (program): Semiprimes in A056105.
- A113523 (program): a(n) = largest composite nonnegative integer <= n.
- A113524 (program): Semiprimes in A056106.
- A113525 (program): Semiprimes in A056107.
- A113526 (program): Define the first two terms to be 1 and 3. All the other terms are obtained by concatenating the two previous terms.
- A113527 (program): Semiprimes in A056108.
- A113528 (program): Semiprimes in A056109.
- A113530 (program): Semiprimes in A003215.
- A113531 (program): a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5.
- A113532 (program): a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6.
- A113536 (program): Numbers k such that k^2 + 13 is prime.
- A113541 (program): Numbers n such that 18n^2+1 is multiple of 19.
- A113549 (program): a(n) = product of n successive numbers up to n, if n is even a(n) = n*(n-1)*.. = n!,if n is odd a(n) = n(n+1)(n+2)… ‘n’ terms.
- A113550 (program): a(n) = product of n successive numbers up to n, if n is odd a(n) = n*(n-1)*.. = n!,if n is even a(n) = n(n+1)(n+2)… ‘n’ terms.
- A113551 (program): a(n) = product of next n even numbers beginning with n if n is even, otherwise product of next n odd numbers beginning with n.
- A113553 (program): Numbers k such that A113552(k) is odd.
- A113555 (program): n-th digit after decimal of the successive approximation of the golden ratio. n-th digit after decimal of F(n+1)/F(n).
- A113574 (program): a(n) is the least n-digit number whose k-th digit is prime if k is prime, composite if k is composite, and 1 if k=1.
- A113580 (program): Define prime(0) = 1; then a(n) = sum prime(d), where d ranges over the decimal digits of n.
- A113581 (program): Define prime(0) = 1; then a(n) = product prime(d), where d ranges over all the decimal digits of n.
- A113582 (program): Triangle T(n,m) read by rows: T(n,m) = (n - m)*(n - m + 1)*m*(m + 1)/4 + 1.
- A113596 (program): P(P(n))-P(P(n-1)), where P(n) = n(n+1)(n+2)/6 (cf. A000292).
- A113597 (program): a(n) = F(F(n+1)) - F(F(n)), where F() = Fibonacci numbers.
- A113601 (program): Intersection of A002144 and A005098.
- A113605 (program): a(1) = a(2) = a(3) = 1; a(n) = a(n-3) + gcd(a(n-1), a(n-2)).
- A113618 (program): a(n) = 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7.
- A113630 (program): 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8.
- A113632 (program): 1 + 2*n + 3*n^2 + 4*n^3 + 5*n^4 + 6*n^5 + 7*n^6 + 8*n^7 + 9*n^8 + 10*n^9.
- A113636 (program): In the sequence of positive integers add 1 to each nonprime number.
- A113637 (program): In the sequence of positive integers subtract 1 from each nonprime number.
- A113638 (program): In the sequence of nonnegative integers subtract 1 from each prime number.
- A113646 (program): a(n) is the smallest composite integer which is >= n.
- A113648 (program): A variant of Josephus Problem in which 2 persons are to be eliminated at the same time.
- A113651 (program): 2*JacobiSymbol(p,5) mod p^2 for p=prime(n).
- A113652 (program): Expansion of (1 - theta_4(q)^2) / 4 in powers of q.
- A113655 (program): Invert blocks of three in the sequence of natural numbers.
- A113657 (program): Decimal expansion of 1/1089.
- A113660 (program): Expansion of phi(x)^3 / phi(x^3) where phi() is a Ramanujan theta function.
- A113661 (program): Expansion of (phi(x)^3/phi(x^3) - 1)/6 where phi() is a Ramanujan theta function.
- A113675 (program): Decimal expansion of 1/8991.
- A113677 (program): a(n) = (2*n+1)!*(2*n-2)!/(2*((n-1)!)*(n!)^2), n=1,2,… .
- A113678 (program): Sequence array for A078008.
- A113679 (program): Expansion of (1-x-2x^2)/(1-x).
- A113680 (program): Riordan array ((1-x-2x^2)/(1-x),x).
- A113681 (program): Expansion of f(-x^2, -x^3)^2 / f(-x, -x^2) in powers of x where f() is Ramanujan’s two-variable theta function.
- A113682 (program): Expansion of 2/(sqrt(1-2*x-3*x^2)*(1+x+sqrt(1-2*x-3*x^2))).
- A113684 (program): Expansion of x(1-x^2-x^3)/((1-x)(1-x-x^2))^2.
- A113687 (program): Expansion of q^(-7/12)eta(q)eta(q^6)^3/(eta(q^2)eta(q^3)) in powers of q.
- A113690 (program): Semiprimes in A054552.
- A113691 (program): Semiprimes in A033951.
- A113692 (program): Semiprimes in A054567.
- A113693 (program): Semiprimes in A054556.
- A113694 (program): Decimal expansion of 10/44955.
- A113697 (program): Floor[n concatenated with n+2 divided by n+1].
- A113704 (program): Triangular indicator function for divisibility, read by rows.
- A113705 (program): Inverse Moebius transform of powers of 10.
- A113709 (program): a(n) is the composite between p(n) and p(n+1), where p(n) is the n-th prime, which is divisible by (p(n+1)-p(n)).
- A113710 (program): a(n) = A113709(n)/(prime(n+1) - prime(n)).
- A113724 (program): A variant of Golomb’s sequence using even numbers: a(n) is the number of times 2*n+2 occurs, starting with a(1) = 2.
- A113726 (program): A Jacobsthal convolution.
- A113727 (program): A Pell convolution.
- A113728 (program): a(n) is the integer between p(n) and p(n+2) which is divisible by (p(n+2)-p(n)), where p(n) is the n-th prime.
- A113742 (program): Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 5 multiples of n-1, n-2, …, 1, for n>=1.
- A113743 (program): Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 6 multiples of n-1, n-2, …, 1.
- A113744 (program): Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 7 multiples of n-1, n-2, …, 1, for n>=1.
- A113745 (program): Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 8 multiples of n-1, n-2, …, 1, for n>=1.
- A113746 (program): Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 9 multiples of n-1, n-2, …, 1, for n>=1.
- A113747 (program): Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 10 multiples of n-1, n-2, …, 1, for n>=1.
- A113748 (program): Generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next 11 multiples of n-1, n-2, …, 1, for n>=1.
- A113753 (program): a(n) = Fibonacci(n-1) + prime(n).
- A113754 (program): Number of possible squares on an n^2 X n^2 grid.
- A113755 (program): Sequence implicit in Jaroma’s corollary to Nagura’s theorem on primes.
- A113763 (program): Non-multiples of 13, or numbers not divisible by 13.
- A113765 (program): Define the first two terms to be 1 and 7. All the other terms are obtained by concatenating the two previous terms.
- A113766 (program): a(n) is the product of those primes which divide some iterate of the Euler totient function but do not divide n itself.
- A113768 (program): a(1) = 1, a(n+1) = a(n) + floor(a(n)^(1/3)).
- A113770 (program): Partial sums of A113311(n)^2.
- A113772 (program): Determinant of the 2 X 2 matrices where the first column is consecutive triangular numbers and the second column is the corresponding consecutive Fibonacci numbers.
- A113773 (program): Number of distinct prime factors of A008352.
- A113774 (program): Number of partitions of {1,…,n} into block sizes not a multiple of 3.
- A113775 (program): Number of sets of lists (cf. A000262) whose list sizes are not a multiple of 3.
- A113778 (program): Invert blocks of four in the sequence of natural numbers.
- A113779 (program): Each term is the sum of the next two digits.
- A113780 (program): Number of solutions to 24*n+1 = x^2+24*y^2, x a positive integer, y an integer.
- A113784 (program): Difference between semiprime(n) and semiprime(n+2).
- A113790 (program): In each block of 5 consecutive natural numbers, swap first and 2nd and swap 4th and 5th.
- A113793 (program): Triangle T(n,m) read by rows: T(n,m) = phi(n - m + 1) * phi(m + 1).
- A113801 (program): Numbers that are congruent to {1, 13} mod 14.
- A113802 (program): Numbers that are congruent to {2, 12} mod 14.
- A113803 (program): Numbers that are congruent to {3, 11} mod 14.
- A113804 (program): Numbers that are congruent to 4 or 10 mod 14.
- A113805 (program): Numbers that are congruent to {5, 9} mod 14.
- A113806 (program): Numbers that are congruent to {6, 8} mod 14.
- A113818 (program): Decimal expansion of the integer (101101101101101101101101101)/9.
- A113823 (program): Tribonacci analog of A055502.
- A113825 (program): Number of distinct prime factors of A008351(n).
- A113828 (program): a(n) = Sum[2^(A047260(i)-1), {i,1,n}].
- A113829 (program): a(n) = a(n-1) + 2^(k(n)), where k(n) is the n-th term of the sequence of numbers that are congruent to {0,3,4,5,7,8} mod 12.
- A113835 (program): a(n) = a(n-1) + 2^(A007494(n-1)).
- A113836 (program): a(n) = Sum[2^(A001651(i-1)-1), {i,1,n}].
- A113837 (program): A number k is included if d(sigma(k)) > sigma(d(k)), where d(k) is number of positive divisors of k and sigma(k) is sum of positive divisors of k.
- A113841 (program): a(n) = a(n-1) + 2^A047240(n) for n>1, a(1)=1.
- A113848 (program): a(1) = a(2) = 1, a(n+2) = 2*a(n) + a(n+1)^2.
- A113849 (program): Numbers whose prime factors are raised to the fourth power.
- A113850 (program): Numbers whose prime factors are raised to the fifth power.
- A113851 (program): Numbers whose prime factors are raised to the sixth power.
- A113852 (program): Numbers whose prime factors are raised to the seventh power.
- A113854 (program): a(n) = sum(2^(A047240(i)-1), i=1..n).
- A113857 (program): a(n) = binomial(4+2*n, n) * binomial(9+2*n, 4+n).
- A113859 (program): Expansion of (7-14*x+6*x^2)/((1-x)*(2*x^2-4*x+1)); related to the binomial transform of Pell numbers A000129 (see formula and comment for A007070).
- A113861 (program): a(n) = (1/9)*((6*n - 7)*2^(n-1) - (-1)^n).
- A113863 (program): Expansion of Molien series for representation Sym^2(R^n) of the automorphism group of the lattice E_8.
- A113865 (program): Number of digits of Bell number A000110(n).
- A113866 (program): Primes in the sequence A064491.
- A113867 (program): a(n) = a(n-1) + 2^(A047258(n)) for n>1, a(1)=1.
- A113870 (program): a(n) = a(n-1) + 2^(k(n)), where k(n) is the n-th term of the sequence formed by k(1)=0 together with the numbers A042963.
- A113871 (program): G.f.: 1/(Sum_{k>=0} (k!)^2 x^k).
- A113873 (program): a(3n) = a(3n-1) + a(3n-2), a(3n+1) = 2n*a(3n) + a(3n-1), a(3n+2) = a(3n+1) + a(3n).
- A113874 (program): a(3n) = a(3n-1) + a(3n-2), a(3n+1) = 2n*a(3n) + a(3n-1), a(3n+2) = a(3n+1) + a(3n).
- A113876 (program): a(n) = a(n-1) + 2^(k(n)), where k(n) is the n-th term of the sequence formed by k(1)=0 together with the numbers A042964.
- A113888 (program): C(2*n+1,n)*C(2*n+6,n+1).
- A113894 (program): a(n) = binomial(2*n, n) * binomial(5+2*n, n).
- A113895 (program): a(n) = C(2+2*n, n) * C(7+2*n, 2+n).
- A113901 (program): Product of omega(n) and bigomega(n) = A001221(n)*A001222(n), where omega(x): number of distinct prime divisors of x. bigomega(x): number of prime divisors of x, counted with multiplicity.
- A113902 (program): Product of omega(n!) and bigomega(n!).
- A113903 (program): Sum of omega(n!) and bigomega(n!).
- A113904 (program): Starting with 1, each number is the previous number plus the product of the index number and the sum of the digits of the previous number.
- A113909 (program): Square table of odd numbers which are neither squares nor one less than squares, read by antidiagonals.
- A113911 (program): Prime numbers not appearing in the nextprime(x^2) sequence A007491.
- A113920 (program): G.f.: (x^3 - x + 1)^3/(x^3*(1 - x)^3).
- A113923 (program): A Farey like level n=2 rational function as a coefficient expansion.
- A113924 (program): a(n) = gcd(A113605(n+1), A113605(n)). Also, for n >= 2, a(n) = A113605(n+2) - A113605(n-1).
- A113925 (program): a(1)=0. a(1)=1. a(n+2) = gcd(a(n+1) + a(n), n).
- A113935 (program): a(n) = prime(n) + 3.
- A113946 (program): Series expansion of Farey rational polynomial based on A112627.
- A113953 (program): A Jacobsthal triangle.
- A113954 (program): Expansion of (1-2x^2)/((1-2x)(1+x)^2).
- A113955 (program): Riordan array (1/((1-4x)c(x)),xc(x)/sqrt(1-4x)), c(x) the g.f. of A000108.
- A113956 (program): Expansion of (1/((1-4x)c(x)))/(1-x^2c(x)/sqrt(1-4x)), c(x) the g.f. of A000108.
- A113957 (program): Sum of the divisors of n which are not divisible by 7.
- A113968 (program): a(0) = 0 and a(n) = (5*(-4)^n + 16*(-1)^n + 9*4^n)/240 for n >= 1.
- A113973 (program): Expansion of phi(x^3)^3/phi(x) where phi() is a Ramanujan theta function.
- A113974 (program): Expansion of (1-phi(x^3)^3/phi(x))/2 where phi() is a Ramanujan theta function.
- A113975 (program): Devil’s Farey: coefficient expansion of a quadratic over quadratic that has 123 roots and a Farey p[1/2]=1 ( correction).
- A113976 (program): a(n) = 4*a(n-1) - 3*a(n-2), for n>3, with a(0) = 14, a(1) = 133, a(2) = 616, and a(3) = 2128.
- A113978 (program): a(n)=Sum(d|n)(10^(n-d)).
- A113979 (program): Number of compositions of n with an even number of 1’s.
- A113980 (program): Number of compositions of n with an odd number of 1’s.
- A113998 (program): Reverse of triangle A051731.
- A113999 (program): a(n) = Sum_{ k, k|n } 10^(k-1).
- A114000 (program): Triangle read by rows: column k has g.f. = Sum_{k>0} x^k/(1-x^(2*k+1)).
- A114001 (program): Rows of A114000 expressed as decimals (a sequence related to the number of divisors of 2n-1).
- A114002 (program): Expansion of x^k(1+x^(k+1))/(1-x^(k+1)).
- A114003 (program): Rows sums of triangle A114002.
- A114004 (program): Inverse of triangle A114002.
- A114005 (program): First column of number triangle A114004.
- A114006 (program): Row sums of number triangle A114004.
- A114010 (program): a(1) = a(2) = 2, Let k(n) = (prime(n) + prime(n+1))/2. Then a(k(n)) = k(n). a(k(n)-i) = prime(n), a(k(n)+i) = prime(n+1) until the next prime occurs.
- A114011 (program): Least multiple of prime(n) ending in digit 1.
- A114013 (program): Least multiple of prime(n) ending in digit 9.
- A114014 (program): Expansion of g.f. (1 + 2*x)^4/((1 + x)*(1 - 16*x^2)).
- A114039 (program): G.f.: 1/Sum_{k>=0} k!*(k!+1)*x^k/2.
- A114040 (program): a(0) = 1, a(1) = 9, a(n) = 6*a(n-1) - a(n-2) - 4.
- A114046 (program): Numbers x such that x^2 - 92*y^2 = 1 for some y.
- A114047 (program): x such that x^2 - 13*y^2 = 1.
- A114048 (program): x-values in the solution to x^2 - 19*y^2 = 1.
- A114049 (program): x such that x^2 - 21*y^2 = 1.
- A114050 (program): x-values in the solution to x^2 - 22*y^2 = 1.
- A114051 (program): x such that x^2 - 23*y^2 = 1.
- A114052 (program): x such that x^2 - 27*y^2 = 1.
- A114054 (program): Decimal expansion of 998998998998998998998998998/9.
- A114059 (program): a(n) = binomial(3+2*n, n) * binomial(8+2*n, 3+n).
- A114090 (program): Sums of 2 distinct nonnegative cubes.
- A114091 (program): Number of partitions of n into parts that are distinct mod 3.
- A114099 (program): Number of partitions of n into parts with digital root = 9.
- A114103 (program): n multiples of n such that a(n) is a multiple of n. The n-th group contains n multiples of n. Arranged sequentially the n-th term is a multiple of n.
- A114104 (program): a(n) = A114103(n)/n.
- A114105 (program): The sum of the next n terms of A114103.
- A114112 (program): a(1)=1, a(2)=2; thereafter a(n) = n+1 if n odd, n-1 if n even.
- A114113 (program): a(n) = sum{k=1 to n} (A114112(k)). (For n>=2, a(n) = sum{k=1 to n} (A014681(k)) =sum{k=1 to n} (A103889(k)).).
- A114114 (program): An invertible partition matrix.
- A114116 (program): 1’s-counting matrix: row sums give number of 1’s in binary expansion of n+1.
- A114117 (program): Inverse of 1’s counting matrix A114116.
- A114118 (program): Number triangle T(n,k)=sum{j=0..n, C(floor((n+k+j)/3),k)C(k,floor((n+k+j)/3))}.
- A114119 (program): Row sums of triangle A114118.
- A114121 (program): Expansion of (sqrt(1 - 4*x) + (1 - 2*x))/(2*(1 - 4*x)).
- A114122 (program): Expansion of (1+x)^2/(1+2x-4x^3-4x^4).
- A114123 (program): Riordan array (1/(1-x), x*(1+x)^2/(1-x)^2).
- A114143 (program): Possible sums of the final scores of completed American football games where both teams score.
- A114144 (program): A variant of the Josephus Problem in which three persons are to be eliminated at the same time.
- A114160 (program): E.g.f. is A(x) = (1-log(B(x)))/B(x), where B(x) = sqrt(1-2*x).
- A114161 (program): E.g.f.: (3-log(1-2*x))/(1-2*x)^(1/2).
- A114162 (program): C(n,k)*Floor((n-k)/2)!.
- A114164 (program): Riordan array (1/(1-2x), x(1-x)/(1-2x)^2).
- A114180 (program): Numbers n with mu(n) = mu(n+1) = mu(n+2).
- A114182 (program): F(4n) - 2n - 1 where F(n) = Fibonacci numbers. Also, the floor of the log base phi of sequence A090162 (phi = (1+Sqrt(5))/2).
- A114185 (program): a(n) = Fibonacci(2*n) - n - 1.
- A114186 (program): Running sums of consecutive integers with all primes set to 2.
- A114188 (program): Riordan array (1/(1-x),x(1+x)/(1-x)^2).
- A114189 (program): Riordan array (1/(1+xc(-2x)), xc(-2x)/(1+xc(-2x)), c(x) the g.f. of A000108.
- A114191 (program): Expansion of 1/(1+x*c(-2*x)), c(x) the g.f. of A000108.
- A114192 (program): Riordan array (1/(1-2x),x/(1-2x)^2).
- A114193 (program): Riordan array (1/(1+2xc(-2x)),xc(-2x)/(1+2xc(-2x)), c(x) the g.f. of A000108.
- A114195 (program): Riordan array (1/(1-3x),x(1-x)/(1-3x)^2).
- A114196 (program): Expansion of (1-3x)/(1-6x+8x^2+x^3).
- A114198 (program): a(n) = Sum_{k=0..n} binomial(n,k)^2*F(k+1).
- A114199 (program): Row sums of a Pascal-Fibonacci triangle.
- A114201 (program): C(1+2*n,1+n)*C(6+2*n,0+n)
- A114203 (program): Row sums of a Pascal-Jacobsthal triangle.
- A114204 (program): Sum {binomial(n,k)^2*J(k+1),k,0,n} with J(n)=A001045(n).
- A114209 (program): Number of permutations of [n] having exactly two fixed points and avoiding the patterns 123 and 231.
- A114210 (program): Number of derangements of [n] avoiding the patterns 123 and 231.
- A114211 (program): a(n) = (5*n^3+12*n^2+n+6)/6.
- A114212 (program): Generalized Gould sequence.
- A114214 (program): Diagonal sums of number triangle A114213.
- A114215 (program): Number of derangements of [n] avoiding the patterns 123, 132 and 213.
- A114216 (program): a(0)=0; thereafter a(n) = largest odd divisor of a(n-1) + prime(n).
- A114219 (program): Number triangle (k-(k-1)*0^(n-k))*[k<=n].
- A114220 (program): a(n) = Sum_{k=0..floor(n/2)} k-(k-1)*0^(n-2k).
- A114226 (program): Row sums of a Pascal-Thue-Morse triangle.
- A114238 (program): a(n) = binomial(2+2*n, 2+n) * binomial(7+2*n, n).
- A114239 (program): a(n) = (n+1)(n+2)^3*(n+3)(n^2 + 4n + 5)/120.
- A114240 (program): a(n) = (n+1)(n+2)^2*(n+3)(7n^2 + 23n + 20)/240.
- A114241 (program): a(n) = (n+1)*(n+2)*(n+3)*(11*n^2 + 29*n + 20)/120.
- A114242 (program): a(n) = (n+1)(n+2)^2*(n+3)^2*(n+4)(2n+5)/720.
- A114243 (program): a(n) = (n+1)*(n+2)^2*(n+3)*(n+4)*(3*n+5)/240.
- A114244 (program): a(n) = (n+1)*(n+2)^2*(n+3)*(7n^2 + 28n + 30)/360.
- A114251 (program): C(3+2*n,3+n)*C(8+2*n,0+n)
- A114252 (program): C(4+2*n,4+n)*C(9+2*n,0+n)
- A114253 (program): a(n) = C(5+2*n,5+n)*C(10+2*n,0+n).
- A114254 (program): Sum of all terms on the two principal diagonals of a 2n+1 X 2n+1 square spiral.
- A114269 (program): Numbers k such that k^2 + 6 is prime.
- A114270 (program): Numbers k such that k^2 + 7 is prime.
- A114271 (program): Numbers k such that k^2 + 8 is prime.
- A114272 (program): Numbers k such that k^2 + 9 is prime.
- A114273 (program): Numbers k such that k^2 + 10 is prime.
- A114274 (program): Numbers k such that k^2 + 11 is prime.
- A114275 (program): Numbers k such that k^2 + 12 is prime.
- A114277 (program): Sum of the lengths of the second ascents in all Dyck paths of semilength n+2.
- A114278 (program): A triangle generated by a family of Sierpinski-triangle related polynomials.
- A114279 (program): Row sums of number triangle A114278.
- A114283 (program): Sequence array for binomial transform of Jacobsthal numbers A001045(n+1).
- A114284 (program): Riordan array ((1-3*x)/(1-x), x).
- A114285 (program): Expansion of (1-3*x)/((1-x)*(1-x^2)).
- A114300 (program): Number of non-intersecting cycle systems in a particular directed graph.
- A114307 (program): Length of the cycle for Lucas numbers mod 10^n.
- A114310 (program): (n-1)!*(n!-n*(n-1)/2).
- A114311 (program): a(n) = n! - n(n-1)/2.
- A114322 (program): Largest number whose 4th power has n digits.
- A114327 (program): Table T(n,m) = n - m read by upwards antidiagonals.
- A114329 (program): Triangle T(n,k) is the number of partitions of an n-set into lists (cf. A000262) with k lists of size 1.
- A114334 (program): Divisors of 6^6.
- A114335 (program): Numbers k such that k^2 + 1 and k^2 - 3 are both prime.
- A114338 (program): Number of divisors of n!! (double factorial = A006882(n)).
- A114347 (program): Cumulative sum of triple factorial numbers a(n) = n!!! (A007661).
- A114350 (program): Primes of the form 2x^3 + x + 1.
- A114351 (program): Primes of the form 3x^3+x+1.
- A114352 (program): Primes of the form 5x^3+x+1.
- A114353 (program): Primes of the form 7x^3+x+1.
- A114354 (program): Primes of the form 9x^3+x+1.
- A114364 (program): a(n) = n*(n+1)^2.
- A114372 (program): Number of partitions of n into parts with an odd number of prime factors that are all distinct.
- A114374 (program): Number of partitions of n into parts that are not squarefree.
- A114375 (program): a(n) = (a(n-1) XOR a(n-2)) + 1, a(0) = a(1) = 0.
- A114378 (program): Area of annuli of consecutive integer thickness.
- A114379 (program): Sums of p-th to the q-th prime where p and q are twin primes.
- A114389 (program): Bisection of A065621.
- A114390 (program): a(n) = A065621(n^2).
- A114398 (program): Positions where A000695 is a square.
- A114399 (program): Squares in A000695.
- A114400 (program): Square roots of A114399.
- A114403 (program): Triprime gaps. First differences of A014612.
- A114404 (program): 4-almost prime gaps. First differences of A014613.
- A114405 (program): 5-almost prime gaps. First differences of A014614.
- A114406 (program): 6-almost prime gaps. First differences of A046306.
- A114407 (program): 7-almost prime gaps. First differences of A046308.
- A114408 (program): 8-almost prime gaps. First differences of A046310.
- A114410 (program): Cumulative sum of double primorials (A079078).
- A114411 (program): Triple primorial n### = n#3.
- A114420 (program): Quadruple primorial n#### = n#4.
- A114421 (program): Quintuple primorial n##### = n#5.
- A114422 (program): Riordan array (1/sqrt(1-2*x-3*x^2), M(x)-1) where M(x) is the g.f. of the Motzkin numbers A001006.
- A114423 (program): Multifactorial array read by ascending antidiagonals.
- A114425 (program): Product of the first n 3-almost primes (A014612).
- A114426 (program): Product of the first n 4-almost primes (A014613).
- A114427 (program): Decimal expansion of the real solution of x^3-x^2-x-4=0.
- A114431 (program): Decimal expansion of the real solution of x^3 - x^2 - 2x - 4 = 0.
- A114433 (program): Last digit (the checksum) of 10-digit ISBN numbers, 10 is represented as “X”.
- A114434 (program): To obtain a(n), write the n-th composite number as a product of primes, add 1 to each prime and multiply.
- A114435 (program): Indices of 4-almost prime triangular numbers.
- A114436 (program): Indices of 5-almost prime triangular numbers.
- A114437 (program): Indices of 6-almost prime triangular numbers.
- A114439 (program): Indices of semiprime pentagonal numbers.
- A114441 (program): Indices of 3-almost prime pentagonal numbers.
- A114443 (program): Indices of 4-almost prime pentagonal numbers.
- A114444 (program): a(n) = 16*n*(n+2).
- A114445 (program): Indices of 5-almost prime pentagonal numbers.
- A114446 (program): Indices of 7-almost prime pentagonal numbers.
- A114447 (program): Indices of 6-almost prime pentagonal numbers.
- A114448 (program): Array a(n,k) = n^k (mod k) read by antidiagonals (k>=1, n>=1).
- A114450 (program): a(n)=(2n)!*(sum{k=1…2n}1/k)/(2n+1).
- A114454 (program): Numbers n such that the n-th hexagonal number is a 3-almost prime.
- A114455 (program): Numbers n such that the n-th hexagonal number is a 4-almost prime.
- A114456 (program): Numbers n such that the n-th hexagonal number is a 5-almost prime.
- A114458 (program): Integer part of sqrt(n)+sqrt(n+1)+sqrt(n+2).
- A114459 (program): Integer part of sqrt(n)+sqrt(n+1)+sqrt(n+2)+sqrt(n+3).
- A114460 (program): Integer part of sqrt(n)+sqrt(n+1)+sqrt(n+2)+sqrt(n+3)+sqrt(n+4).
- A114464 (program): Number of Dyck paths of semilength n having no ascents of length 2 that start at an even level.
- A114465 (program): Number of Dyck paths of semilength n having no ascents of length 2 that start at an odd level.
- A114476 (program): Triangle read by rows: inverse of triangle in A061554 with signs in each column +,+,-,-,+,+,-,-,…
- A114479 (program): Kekulé numbers for certain benzenoids.
- A114480 (program): Kekulé numbers for certain benzenoids.
- A114481 (program): Kekulé numbers for certain benzenoids.
- A114482 (program): Let S(1)=1, S(2)=10; S(2n)=concatenation of S(2n-1), S(2n-2) and 0; and S(2n+1)=concatenation of S(2n), S(2n) and 0. Sequence gives S(infinity).
- A114487 (program): Number of Dyck paths of semilength n having no UUDD’s starting at level 0.
- A114494 (program): Triangle read by rows: T(n,k) is number of hill-free Dyck paths of semilength n and having k returns to the x-axis. (A Dyck path is said to be hill-free if it has no peaks at level 1.)
- A114495 (program): Number of returns to the x-axis in all hill-free Dyck paths of semilength n (a Dyck path is said to be hill-free if it has no peaks at level 1).
- A114496 (program): a(n) = Sum of binomial(n,k)*binomial(2n+k,k) over all k.
- A114504 (program): Numbers n such that the n-th hexagonal number is a 6-almost prime.
- A114505 (program): Numbers n such that the n-th hexagonal number is a 7-almost prime.
- A114507 (program): Number of Dyck paths of semilength n having no ascents of length 3.
- A114509 (program): Number of Dyck paths of semilength n having no ascents of length 4.
- A114514 (program): The digits on a numerical pad from upper left to lower right.
- A114515 (program): Number of peaks in all hill-free Dyck paths of semilength n (a Dyck path is hill-free if it has no peaks at level 1).
- A114517 (program): Numbers n such that n-th heptagonal number is semiprime.
- A114518 (program): Numbers n such that A008475(n) is prime.
- A114519 (program): a(n) = A008475(A114518(n)).
- A114520 (program): Composites in sequence A114518.
- A114521 (program): a(n) = A008475(A114520(n)).
- A114522 (program): Numbers n such that sum of distinct prime divisors of n is prime.
- A114525 (program): Triangle of coefficients of the Lucas (w-)polynomials.
- A114540 (program): Number of correct decimal digits given by the n-th convergent to the golden ratio.
- A114543 (program): Expansion of x*(1+x)/((1+2x)(1-2x-768x^2)).
- A114548 (program): Numbers n such that n-th heptagonal number is 3-almost prime.
- A114551 (program): Continued fraction expansion of the constant (A114550) equal to Sum_{n>=0} 1/A112373(n) such that the partial quotients satisfy a(2n) = A112373(n) for n > 0 and a(2n+1) = A112373(n+1)/A112373(n) for n >= 0.
- A114553 (program): a(n) = 25*a(n-2) + 48*a(n-3) with a(0) = 0, a(1) = a(2) = 1.
- A114554 (program): Numbers n such that n-th heptagonal number is 4-almost prime.
- A114555 (program): Smallest nonsquarefree integer > the n-th term of the Fibonacci sequence.
- A114556 (program): Numbers n such that n-th heptagonal number is 5-almost prime.
- A114558 (program): Numbers n such that n-th heptagonal number is 6-almost prime.
- A114559 (program): Numbers n such that n-th heptagonal number is 7-almost prime.
- A114562 (program): The first occurrence of n in A111701.
- A114564 (program): Numbers of the form n=12s+7, where q=4s+3 is a prime for which the order of 2 is either q-1 or (q-1)/2.
- A114567 (program): Numbers k such that the binary expansion of n mod 2^k is the postorder traversal of a binary tree, where 1 indicates a node and 0 indicates there are no children on that side.
- A114569 (program): a(n) = 9*4^n - 1.
- A114570 (program): Let the decimal expansion of n be d_1 d_2 … d_k; then a(n) = Sum_{i=1..k} d_i^(k+1-i).
- A114574 (program): a(n) = p*p! where p = prime(n).
- A114587 (program): Number of peaks at odd levels in all hill-free Dyck paths of semilength n+3 (a hill in a Dyck path is a peak at level 1).
- A114589 (program): Number of hill-free Dyck paths of semilength n+3 and having no peaks at even levels (a hill in a Dyck path is a peak at level 1).
- A114590 (program): Number of peaks at even levels in all hill-free Dyck paths of semilength n+2 (a hill in a Dyck path is a peak at level 1).
- A114604 (program): Numerator of partial sums of A005329/A006125.
- A114606 (program): Numbers n such that n-th octagonal number is 3-almost prime.
- A114607 (program): Start with 1 0 1 0 then add a one every time (e.g. 1 1 0 1 1 1 0 1 1 1 1 0 …).
- A114618 (program): Numbers n such that n-th octagonal number is 4-almost prime.
- A114619 (program): a(n) = 2*A079291(n) (twice squares of Pell numbers).
- A114620 (program): 2*A084158 (twice Pell triangles).
- A114621 (program): Numbers n such that n-th octagonal number is 5-almost prime.
- A114633 (program): a(n) = (n+1)*(n+2)/2 * Sum_{k=0..floor(n/2)} n!/(n-2*k)!.
- A114634 (program): Numbers n such that n-th octagonal number is 6-almost prime.
- A114635 (program): Numbers n such that n-th octagonal number is 7-almost prime.
- A114636 (program): Numbers n such that n-th octagonal number is 8-almost prime.
- A114637 (program): Nonnegative numbers excluding 1 and 2.
- A114643 (program): Number of real primitive Dirichlet characters modulo n.
- A114646 (program): Let M(n) be the n X n matrix m(i,j)=min(i,j) for 1<=i,j<=n then a(n) is the trace of M(n)^(-4).
- A114647 (program): Expansion of (3 -4*x -3*x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.
- A114652 (program): a(1)=1. For n>1, a(n) = a(n-1) + (number of terms among {a(1),a(2),…,a(n-1)} which are coprime to n).
- A114653 (program): A114652(n+1) - A114652(n).
- A114654 (program): Discriminant of the polynomial x^n + x + 1.
- A114655 (program): Triangle read by rows: T(n,k) is the number of Schroeder paths of length 2n and having k weak ascents (1<=k<=n). A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A weak ascent in a Schroeder path is a maximal sequence of consecutive U and H steps.
- A114656 (program): Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k peaks (1 <= k <= n).
- A114687 (program): Triangle read by rows: T(n,k) is the number of double rise-bicolored Dyck paths (double rises come in two colors; also called marked Dyck paths) of semilength n and having k double rises (0 <= k <= n-1).
- A114688 (program): Expansion of (1 +3*x -x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.
- A114689 (program): Expansion of (1 +4*x -x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.
- A114693 (program): Number of returns to the x-axis in all hill-free Schroeder paths of length 2n+4. A Schroeder path of length 2n is a lattice path from (0,0) to (2n,0) consisting of U=(1,1), D=(1,-1) and H=(2,0) steps and never going below the x-axis. A hill is a peak at height 1.
- A114696 (program): Expansion of (1+4*x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.
- A114697 (program): Expansion of (1+x+x^2)/((1-x^2)*(1-2*x-x^2)); a Pellian-related sequence.
- A114698 (program): Let M(n) be the n X n matrix m(i,j)=min(i,j) for 1<=i,j<=n then a(n) is the trace of M(n)^(-3).
- A114710 (program): Number of hill-free Schroeder paths of length 2n that have no horizontal steps on the x-axis.
- A114713 (program): Number of ascents in all peakless Motzkin paths of length n+3.
- A114723 (program): G.f.: x*(1 - 2*x^2)/(1 - x - 3*x^2 - 3*x^3 - x^4).
- A114743 (program): a(1) =4, a(2) = 6, a(n+1) = least composite number of the form k*(a(n-1)) - a(n), not included earlier.
- A114745 (program): a(1) = 1, a(2) = 3, a(n+1) = least number of the form k*(a(n-1)) - a(n), not included earlier.
- A114751 (program): The following triangle contains n consecutive numbers beginning from n in ascending order if n is odd else in descending order. Sequence contains the triangle by rows.
- A114752 (program): a(2n)=2n, a(2n+1)=4n+1.
- A114753 (program): First column of A114751.
- A114760 (program): n-th digit after decimal point of the successive approximations to phi = .618…; equally, n-th digit after decimal point of F(n)/F(n+1).
- A114761 (program): a(n) = (floor(sqrt(2)*10^n))^2.
- A114762 (program): a(n) = floor(3^(1/2)*10^n)^2.
- A114763 (program): a(n) = floor(sqrt(5)*10^n)^2.
- A114764 (program): a(n) = floor(sqrt(6)*10^n)^2.
- A114765 (program): a(n) = floor(sqrt(7) * 10^n)^2.
- A114766 (program): a(n) = floor(sqrt(8)*10^n)^2.
- A114767 (program): Floor[2^(1/3)*10^n]^3.
- A114768 (program): a(n) = floor(3^(1/3)*10^n)^3.
- A114769 (program): a(n) = floor(4^(1/3)*10^n)^3.
- A114770 (program): Floor[5^(1/3)*10^n]^3.
- A114771 (program): Floor[6^(1/3)*10^n]^3.
- A114772 (program): Floor[7^(1/3)*10^n]^3.
- A114773 (program): Floor[9^(1/3)*10^n]^3.
- A114775 (program): Expansion of x^2*(1+x^2)*(1 - x^4 + x^7)/((1 - x^4 + x^6)*(1 - x^4 - x^6)).
- A114778 (program): Cumulative product of triple factorial A007661.
- A114779 (program): Cumulative product of quadruple factorial A007662.
- A114790 (program): Cumulative product of quintuple factorial A085157.
- A114793 (program): a(1) = a(2) = 1; for n>2, a(n) = a(n-2)^3 + a(n-1)^2.
- A114795 (program): {concatenation n, n-1, n-2, …3,2,1} mod n.
- A114796 (program): Cumulative product of sextuple factorial A085158.
- A114797 (program): a(n) = n multiplied by the least nontrivial nondivisor of n.
- A114798 (program): Cubic polynomial coefficients such that an elliptical term is zero.
- A114799 (program): Septuple factorial, 7-factorial, n!7, n!!!!!!!, a(n) = n*a(n-7) if n > 1, else 1.
- A114800 (program): Octuple factorial, 8-factorial, n!8, n!!!!!!!!.
- A114803 (program): Integers when g2^3-27*g3^2=0 in cubic polynomials of the form: w^2=4*x^3-g2*x-g3.
- A114805 (program): Cumulative sum of quintuple factorial numbers n!!!!! (A085157).
- A114806 (program): Nonuple factorial, 9-factorial, n!9, n!!!!!!!!!.
- A114810 (program): Number of complex, weakly primitive Dirichlet characters modulo n.
- A114811 (program): Number of real, weakly primitive Dirichlet characters modulo n.
- A114828 (program): Numbers n such that n-th octagonal number is 9-almost prime.
- A114831 (program): Each term is previous term plus floor of harmonic mean of two previous terms.
- A114832 (program): Each term is previous term plus ceiling of harmonic mean of two previous terms.
- A114846 (program): Numbers of the form p^p - p!, where p is a prime.
- A114853 (program): a(n) = floor(n^n/n!!).
- A114855 (program): Expansion of q^(-1/3) * (eta(q) * eta(q^4))^2 / eta(q^2) in powers of q.
- A114870 (program): a(n) = A002627(n+1) - A002627(n) - n!.
- A114872 (program): Even numbers not representable as (p-1)p^k (where p is a prime and k>=0) in ascending order.
- A114873 (program): Numbers representable in exactly one way as (p-1)p^k (where p is a prime and k>=0), in ascending order.
- A114889 (program): a(1)=1 and, for n>1, a(n) is the smallest integer greater than a(n-1) such that a(n)+a(i) is not a power of 3, for i=1,…, n-1.
- A114890 (program): First differences of A114889.
- A114891 (program): Numbers that are the smallest element of a k-cycle (k > 1) of permutation A114861.
- A114892 (program): a(n) is the cycle length corresponding to A114891(n).
- A114900 (program): Number of compositions of n such that no two adjacent parts are equal, allowing 0.
- A114912 (program): 2^a(n) divides A000009(n) but 2^(a(n)+1) does not.
- A114913 (program): Numbers n such that A114912(n)=1. Numbers n such that A000009(n) == 2 (mod 4).
- A114914 (program): Values in A114913 that are not in A111174.
- A114938 (program): Number of permutations of the multiset {1,1,2,2,…,n,n} with no two consecutive terms equal.
- A114945 (program): Number of monic irreducible polynomials over GF(3) of degree <= n.
- A114946 (program): Number of monic irreducible polynomials over GF(4) of degree <= n.
- A114947 (program): Number of monic irreducible polynomials over GF(5) of degree <= n.
- A114948 (program): a(n) = n^2 + 10.
- A114949 (program): a(n) = n^2 + 6.
- A114955 (program): A 2/3-power Fibonacci sequence.
- A114956 (program): a(0) = a(1) = 1, for n>1 a(n) = ceiling(a(n-1)^(3/4) + a(n-2)^(3/4)).
- A114958 (program): a(n) = 6*2^(n+1) - 5*(n+1) - 4.
- A114960 (program): Expansion of (-1+3*x-5*x^2+4*x^3) / ((1-2*x)*(2*x^2-1)*(x-1)^2).
- A114962 (program): a(n) = n^2 + 14.
- A114963 (program): a(n) = n^2 + 22.
- A114964 (program): a(n) = n^2 + 30.
- A114965 (program): n^2 + 34.
- A114982 (program): Expansion of x(3-x^2)/(1-3x).
- A114984 (program): Coefficients of cubic equations in the form w^2=4*x^3-g2*x-g3 Weierstrass elliptic form whose solutions approximate zeta zeros.
- A114986 (program): Characteristic function of (A000201 prefixed with 0).
- A114989 (program): Numbers whose sum of squares of distinct prime factors is prime.
- A114990 (program): a(n) = a(n-2) + A000265(a(n-1)), a(0)=0, a(1)=1.
- A114994 (program): Numbers whose binary representation has monotonically decreasing sizes of groups of zeros (including zero-length groups between adjacent ones).
- A114997 (program): Number of ordered trees with n edges and no unary or binary nodes.
- A115000 (program): a(n) = J_2(n) / 24.
- A115001 (program): Sum_{k=1}^n J_2(k)/24.
- A115002 (program): J_4(n)/240.
- A115003 (program): Sum_{k=1}^n J_4(k)/240.
- A115006 (program): Row 2 of array in A114999.
- A115007 (program): Row 3 of array in A114999.
- A115008 (program): a(n) = a(n-1)+a(n-3)+a(n-4).
- A115012 (program): Sum_{i=1..n, gcd(5,i)=1} i.
- A115013 (program): a(n) = difference between largest and smallest primes dividing the n-th squarefree integer (with a(1)=0).
- A115014 (program): Sum_{i=1..n, gcd(6,i)=1} i.
- A115015 (program): Sum_{i=1..n} (gcd(7,i)=1) i.
- A115017 (program): a(n) = largest triangular number dividing n.
- A115018 (program): Numbers n such that (n+1)*(n+2)^2 + 1 is prime.
- A115020 (program): Count backwards from 100 in steps of 7.
- A115022 (program): a(n) = F(n-th squarefree)/product{p=primes,p|(n-th squarefree)} F(p), where F(m) is m-th Fibonacci number.
- A115024 (program): Natural numbers n such that the number of prime factors of n (counted with multiplicity) is equal to the number of decimal digits of n.
- A115025 (program): a(n) = n-th element of n-th row of triangle shown below.
- A115030 (program): Number of distinct sums of subsets of the first n prime numbers.
- A115032 (program): Expansion of (5-14*x+x^2)/((1-x)*(x^2-18*x+1)).
- A115034 (program): Alternately multiply and divide, with a(1)=1 and a(2)=2.
- A115036 (program): Even terms of A116883.
- A115048 (program): Count backwards from 100 in steps modulo n.
- A115052 (program): Expansion of 1/(3*x^2 - 3*x + 1)^2.
- A115053 (program): Series expansion of x*(x+3)^2/(3*x+1)^2.
- A115055 (program): Lower level digraph derived from a voltage graph.
- A115056 (program): a(n) = n*(n^2-1)*(3*n+2).
- A115058 (program): Primes p such that 3p+2 is not prime.
- A115059 (program): a(n+4) = a(n+3)+a(n+1)+a(n)+k(n), where k(n) = 0, 1, 0, or -1 according to n mod 4.
- A115061 (program): a(n) is the number of occurrences of the n-th prime number in A051697.
- A115065 (program): Number of points with integer coordinates inside the equilateral triangle with base [0,n].
- A115066 (program): Chebyshev polynomial of the first kind T(n,x), evaluated at x=n.
- A115067 (program): a(n) = (3*n^2 - n - 2)/2.
- A115068 (program): Triangle read by rows: T(n,k) = number of elements in the Coxeter group D_n with descent set contained in {s_k}, for 0<=k<=n-1.
- A115069 (program): a(n) = 3^b(n), where b(n) is #{primes p=1 mod 3 dividing n}.
- A115070 (program): a(n) = phi(n)/3^b(n), where b(n) is #{primes p=1 mod 3 dividing n}.
- A115074 (program): a(n) is the largest prime dividing the n-th nonsquarefree positive integer.
- A115079 (program): Matrix log of triangle A051731, where nonzero elements in the matrix log are all unit fractions and represented here by the denominators, with zero elements remaining zero.
- A115081 (program): Column 0 of triangle A115080.
- A115082 (program): Column 1 of triangle A115080.
- A115090 (program): a(n) = A115074(n) - A117183(n).
- A115092 (program): The number of m such that prime(n) divides m!+1.
- A115093 (program): Primes of the form p*q-2, where p and q are distinct primes.
- A115098 (program): a(0)=2, a(n)=3*a(n-1)-3.
- A115099 (program): a(0)=4, a(n) = 3*a(n-1) - 4.
- A115102 (program): a(0)=2, a(1)=8, a(n)=a(n-1)+2*a(n-2).
- A115104 (program): Numbers n such that 4*n^3 + 1 is prime.
- A115107 (program): Numerator of q_n = -4*n + 2*(1+n)*HarmonicNumber(n).
- A115110 (program): Expansion of q^(-1/24) * eta(q)^3 / eta(q^2) in powers of q.
- A115112 (program): Number of different ways to select n elements from two sets of n elements under the precondition of choosing at least one element from each set.
- A115113 (program): a(n) = 3*a(n-1) + 4*a(n-2), with a(0) = 2, a(1) = 6, a(2) = 10.
- A115114 (program): Asymmetric rhythm cycles (patterns): binary necklaces of length 2n subject to the restriction that for any k if the k-th bead is of color 1 then the (k+n)-th bead (modulo 2n) is of color 0.
- A115115 (program): Number of 3-asymmetric rhythm cycles: binary necklaces of length 3n subject to the restriction that for any k if the k-th bead is of color 1 then the (k+n)-th and (k+2n)-th beads (modulo 3n) are of color 0.
- A115125 (program): A sequence related to Catalan numbers A000108.
- A115126 (program): First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).
- A115128 (program): Row sums of triangle A115127.
- A115129 (program): Partial sums of A005587. Fourth column of triangle A115127.
- A115130 (program): Partial sums of A005557.
- A115132 (program): Partial sums of A064059.
- A115133 (program): Partial sums of A064061.
- A115134 (program): Third diagonal sequence of triangle A115127.
- A115136 (program): Row sums of triangle A113647.
- A115137 (program): Second diagonal of triangle A113647 (called Y(2,1)).
- A115138 (program): A sequence related to Catalan numbers A000108.
- A115139 (program): Array of coefficients of polynomials related to integer powers of the generating function of Catalan numbers A000108.
- A115140 (program): O.g.f. inverse of Catalan A000108 o.g.f.
- A115141 (program): Convolution of A115140 with itself.
- A115142 (program): Third convolution of A115140.
- A115143 (program): a(n) = -4*binomial(2*n-5, n-4)/n for n > 0 and a(0) = 1.
- A115144 (program): Fifth convolution of A115140.
- A115145 (program): Sixth convolution of A115140.
- A115146 (program): Seventh convolution of A115140.
- A115147 (program): Eighth convolution of A115140.
- A115148 (program): Ninth convolution of A115140.
- A115149 (program): Tenth convolution of A115140.
- A115150 (program): Third diagonal (M=3) sequence of triangle A113647, called Y(2,1).
- A115158 (program): Number of divisors of A006558(n).
- A115162 (program): Positive numbers that are not the sum of a triangular number, a square and a cube, all of them greater than or equal to 1.
- A115164 (program): a(n) = 3*a(n-1) + 4*a(n-2), with a(0) = 3, a(1) = 7, a(3) = 9, for n > 2.
- A115165 (program): Odd numbers k such that k-1 and k+1 have the same number of distinct prime divisors.
- A115166 (program): Even numbers k such that k-2 and k+2 have the same number of distinct prime factors.
- A115167 (program): Odd numbers k such that k-1 and k+1 have the same number of prime divisors with multiplicity.
- A115178 (program): Expansion of c(x^2+x^3), c(x) the g.f. of A000108.
- A115179 (program): Expansion of c(x*y*(1-x)), c(x) the g.f. of A000108.
- A115180 (program): Beatty sequence for (Champernowne constant)*10 = 1.234567891011121314….
- A115181 (program): Beatty sequence for (Cc/(Cc-1)) with Cc = 1.234567891011121314… = 10*(Champernowne constant).
- A115193 (program): Generalized Catalan triangle of Riordan type, called C(1,2).
- A115194 (program): A sequence related to A000108 (Catalan numbers).
- A115197 (program): Convolution of generalized Catalan numbers A064062 (called C(n;2)).
- A115202 (program): Fifth column of triangle A115193 (called C(1,2)).
- A115203 (program): Sixth column of triangle A115193 (called C(1,2)).
- A115204 (program): Seventh column of triangle A115193 (called C(1,2)).
- A115205 (program): a(n) = binomial(n, 9) + 1.
- A115216 (program): “Correlation triangle” for 2^n.
- A115217 (program): Diagonal sums of “correlation triangle” for 2^n.
- A115218 (program): Triangle read by rows: zeroth row is 0; to get row n >= 1, append next 2^n numbers to end of previous row.
- A115219 (program): Expansion of 2*x^2*(1-x)/(1-3*x+2*x^2-2*x^3).
- A115224 (program): Number of 3 X 3 symmetric matrices over Z(n) having determinant 1.
- A115226 (program): Order of the group of invertible 3 X 3 symmetric matrices over Z(n).
- A115228 (program): Nonsquarefree numbers n such that 2n+1 is also nonsquarefree (A013929).
- A115232 (program): Primes p which can be written in the form 2^i + q^j where i >= 0, j >= 1, q = odd prime.
- A115235 (program): Expansion of eta(q)^2*eta(q^9)*eta(q^18)/(eta(q^2)*eta(q^3)) in powers of q.
- A115238 (program): Row sums of triangle A115237.
- A115241 (program): Square array read by antidiagonals: T(n,p) is the number of linearly independent, homogeneous harmonic polynomials of degree n in p variables (n,p>=1).
- A115243 (program): G.f.: (4*x^2 + 2*x)/(4*x^3 - x^2 - 4*x + 1).
- A115246 (program): Number of different ways to select n elements from three sets of n elements such that there is at least one element from each set.
- A115247 (program): 2^a(n) divides A001935(n) but 2^(a(n)+1) does not.
- A115248 (program): Values such that A115247(a(n))=1. Values such that A001935(a(n))==2 (mod 4).
- A115255 (program): “Correlation triangle” of central binomial coefficients A000984.
- A115256 (program): Diagonal sums of correlation triangle of central binomial coefficients.
- A115257 (program): Partial sums of binomial(2n,n)^2.
- A115259 (program): Difference between the sum of digits in odd positions and the sum of digits in even positions of prime numbers.
- A115262 (program): Correlation triangle for n+1.
- A115264 (program): Diagonal sums of correlation triangle for floor((n+2)/2).
- A115266 (program): Row sums of correlation triangle for floor((n+3)/3).
- A115269 (program): Row sums of correlation triangle for floor((n+4)/4).
- A115270 (program): Diagonal sums of correlation triangle for floor((n+4)/4).
- A115271 (program): Partial sums of floor((n+4)/4)^2.
- A115273 (program): Floor(n/3)*(n mod 3).
- A115274 (program): a(n) = n + A115273(n), where A115273(n) = 0 for n = 1..3.
- A115281 (program): Correlation triangle for the sequence 2-0^n.
- A115282 (program): Correlation triangle for the sequence 3-2*0^n.
- A115283 (program): Diagonal sums of correlation triangle for 3-2*0^n.
- A115285 (program): Diagonal sums of correlation triangle for 1,3,4,4,4,…(A113311).
- A115286 (program): a(n) = (1/6)*(n^6+3*n^4+12*n^3+8*n^2).
- A115291 (program): Expansion of (1+x)^3/(1-x).
- A115293 (program): Row sums of correlation triangle for (1+x)^3/(1-x).
- A115294 (program): Diagonal sums of correlation triangle for (1+x)^3/(1-x).
- A115295 (program): Partial sums of squares of A115291(n).
- A115296 (program): Skew version of correlation triangle for constant sequence 1.
- A115297 (program): Treillis triangle: a triangle read by rows showing the coefficients of sum formulas of Treillis numbers (A115298). The k-th row (k>=1) contains a(n,k) for n=1 to (k+1)/2 (odd rows) and for n=1 to k/2 (even rows), where a(n,k) satisfies Sum_{n=1..[(k+1)/2_odd, k/2_even]} a(n,k). The last term of each row (and its only odd number) equals Prime(k+1)-2.
- A115299 (program): Greatest digit of n + least digit of n. Different from A088133.
- A115300 (program): Greatest digit of n * least digit of n.
- A115302 (program): Permutation of natural numbers generated by 3-rowed array shown below.
- A115303 (program): a(n) = if n<3 then n else 3*a(floor(n/3)) + 2 - n mod 3.
- A115305 (program): a(n) = if n<5 then n else 5*a(floor(n/5)) + 4 - n mod 5.
- A115309 (program): a(n) = if n<9 then n else 9*a(floor(n/9)) + 8 - n mod 9.
- A115311 (program): a(n) = gcd(Lucas(n)-1, Fibonacci(n)-1).
- A115312 (program): a(n) = gcd(Lucas(n)-1, Fibonacci(n)+1).
- A115313 (program): a(n) = gcd(Lucas(n)+1, Fibonacci(n)+1).
- A115314 (program): a(n) = gcd(Lucas(n)+1, Fibonacci(n)-1).
- A115315 (program): a(n) = floor(L^3*{phi^(3*n-2), phi^(3*n-1), phi^(3*n-2) + phi^(3*n-1)}) where L = (1 + sqrt(5))/(2*sqrt(5)) and phi = (1 + sqrt(5))/2.
- A115318 (program): Inverse of A115316.
- A115319 (program): a(n) = A115318(A115318(n)).
- A115322 (program): Triangle of coefficients of Pell polynomials.
- A115326 (program): E.g.f.: exp(x/(1-2*x))/sqrt(1-4*x^2).
- A115327 (program): E.g.f.: exp(x + 3/2*x^2).
- A115328 (program): E.g.f: exp(x/(1-3*x))/sqrt(1-9*x^2).
- A115329 (program): E.g.f.: exp(x + 2*x^2).
- A115330 (program): E.g.f: exp(x/(1-4*x))/sqrt(1-16*x^2).
- A115331 (program): E.g.f.: exp(x+5/2*x^2).
- A115332 (program): E.g.f: exp(x/(1-5*x))/sqrt(1-25*x^2).
- A115333 (program): Sum of primes that do not divide n and are less than the largest prime dividing n.
- A115334 (program): Numbers d > 0 such that 3+2d and 3+4d are primes.
- A115335 (program): a(0) = 3, a(1) = 5, a(2) = 1, and a(n) = (2^(1 + n) - 11*(-1)^n)/3 for n > 2.
- A115338 (program): a(n)=F([sqrt(n)]), where [k]=integer part of k and F(n) is the Fibonacci sequence.
- A115339 (program): a(2n-1)=F(n+1), a(2n)=L(n), where F(n) and L(n) are the Fibonacci and the Lucas sequences.
- A115341 (program): a(n) = abs(A154879(n+1)).
- A115342 (program): 1 + (n-6)*2^(n-1).
- A115346 (program): Triangle read by rows: T(n,k) = Fibonacci(k+2) - 1.
- A115350 (program): Termination of the aliquot sequence starting at n.
- A115352 (program): Concatenation of finite strings S_0, S_1, S_2, …, where S_0 = {0} and for k >= 1, S_k is obtained from S_{k-1} by inserting the numbers 2^(k-1) through 2^k-1 after the initial 0.
- A115356 (program): Matrix (1,x)+(x,x^2) in Riordan array notation.
- A115357 (program): Period 6: repeat [1,1,1,0,2,0].
- A115358 (program): Inverse of matrix (1,x)+(x,x^2) (expressed in Riordan array notation).
- A115359 (program): Matrix (1,x)-(x,x^2) in Riordan array notation.
- A115360 (program): Period 6: repeat [1,-1,1,0,0,0].
- A115361 (program): Inverse of matrix (1,x)-(x,x^2) (expressed in Riordan array notation).
- A115362 (program): Row sums of ((1,x) + (x,x^2))^(-1)*((1,x)-(x,x^2))^(-1) (using Riordan array notation).
- A115363 (program): ((1,x)-(x,x^2))^(-2) (using Riordan array notation).
- A115364 (program): a(n) = A000217(A001511(n)), where A001511 is one more than the 2-adic valuation of n, and A000217(n) is the n-th triangular number, binomial(n+1, 2).
- A115367 (program): Row sums of correlation triangle for Fredholm-Rueppel sequence A036987.
- A115375 (program): <h[d,d],s[d,d]*s[d,d]*s[d,d]> where h[d,d] is a homogeneous symmetric function, s[d,d] is a Schur function indexed by two parts, * represents the Kronecker product and <, > is the standard scalar product on symmetric functions.
- A115376 (program): <h[d+1,d-1],s[d,d]*s[d,d]*s[d,d]> where h[d+1,d-1] is a homogeneous symmetric function, s[d,d] is a Schur function indexed by two parts, * represents the Kronecker product and <, > is the standard scalar product on symmetric functions.
- A115378 (program): a(n) = number of positive integers k < n such that n XOR k = (n+k).
- A115379 (program): Number of positive integers k < n such that n XOR k < n and gcd(n,k) is odd.
- A115383 (program): Row sums of Thue-Morse correlation triangle A115382.
- A115384 (program): Partial sums of Thue-Morse numbers A010060.
- A115388 (program): Numerator of rational part of raw moment n of the line point picking problem.
- A115389 (program): Denominator of rational part of raw moment n of the line point picking problem.
- A115390 (program): Binomial transform of tribonacci sequence A000073.
- A115391 (program): a(0)=0; then a(4*k+1)=a(4*k)+(4*k+1)^2, a(4*k+2)=a(4*k+1)+(4*k+3)^2, a(4*k+3)=a(4*k+2)+(4*k+2)^2, a(4*k+4)=a(4*k+3)+(4*k+4)^2.
- A115392 (program): First appearance of n-th prime as prime factor in list of semiprimes.
- A115399 (program): Expansion of c(x^2-x^3), c(x) the g.f. of A000108.
- A115400 (program): Number of n-colorings of the octahedral graph.
- A115402 (program): Difference between 3-almostprime(n) and 3-almostprime(n+3).
- A115403 (program): Numbers n such that n^3+1 is 3-almost prime (product of three primes).
- A115405 (program): Numbers n such that n^k is deficient for all k>0.
- A115411 (program): a(n) = least k such that semiprime(n) divides k-th triangular number.
- A115412 (program): G.f.: (x - 1)/(x^5 - x^3 - x^2 - x - 1).
- A115413 (program): Expansion of (x - 1)/(1 - x^2 + x^3 + x^4 - x^5).
- A115415 (program): Real part of (n + i)^n, with i=sqrt(-1).
- A115416 (program): Imaginary part of (n + i)^n, with i=sqrt(-1).
- A115419 (program): Numbers having a 1 in position 3 of their binary expansion.
- A115420 (program): Numbers having a 1 in position 4 of their binary expansion.
- A115421 (program): Numbers having a 1 in position 5 of their binary expansion.
- A115425 (program): The first four terms of the sequence are doubled, then those numbers are tripled and then those numbers are quadrupled, etc.
- A115450 (program): Number triangle (1/((1-x)(1-2x)),-x)-(x/((1-x)(1-2x)),-x^2) (expressed in the notation of Riordan arrays).
- A115451 (program): Expansion of 1/((1+x)*(1-2*x)*(1+x^2)).
- A115454 (program): Composite positive integers written in base 2.
- A115473 (program): Number of monic irreducible polynomials of degree 2 in GF(2)[x1,…,xn].
- A115489 (program): Number of monic irreducible polynomials of degree 3 in GF(2^n)[x].
- A115490 (program): Number of monic irreducible polynomials of degree 4 in GF(2^n)[x].
- A115491 (program): Number of monic irreducible polynomials of degree 5 in GF(2^n)[x].
- A115492 (program): Number of monic irreducible polynomials of degree 2 in GF(2^n)[x,y].
- A115500 (program): Number of monic irreducible polynomials of degree 1 in GF(2^n)[x1,x2,x3,x4].
- A115504 (program): Number of monic irreducible polynomials of degree 1 in GF(2^n)[x1,x2,x3,x4,x5].
- A115512 (program): Number triangle (1,x)+(x,x^3) expressed in terms of Riordan arrays.
- A115514 (program): Triangle read by rows: row n >= 1 lists first n positive members of A004526 (integers repeated) in decreasing order.
- A115516 (program): The mode of the bits of n (using 0 if bimodal).
- A115517 (program): The mode of the bits of n (using 1 if bimodal).
- A115519 (program): n*(1+3*n+6*n^2)/2.
- A115524 (program): Number triangle (1,-x)+(x,x)/2+(x,-x)/2-(x^2,x^2) (expressed using the notation of stretched Riordan arrays).
- A115525 (program): Periodic {1,1,-2,0,1,0,-1,0,0,1,-1,-1}.
- A115526 (program): Inverse of number triangle A115524.
- A115535 (program): Numbers k such that the concatenation of k with 4*k gives a square.
- A115536 (program): Numbers k such that the square of k is the concatenation of two numbers m and 4*m.
- A115561 (program): a(n) = lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.
- A115562 (program): a(n) = number of distinct squarefree ternary (cyclic) sequences uniquely containing every possible length-n substring.
- A115565 (program): a(n) = 5*n^4 - 10*n^3 + 20*n^2 - 15*n + 11.
- A115566 (program): Numbers k such that 2^k, 2^(k+1) and 2^(k+2) have the same number of digits.
- A115567 (program): a(n) = C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).
- A115568 (program): Maximal Fibonacci exponent in prime factorization of n, or 1 if there is no Fibonacci exponent.
- A115588 (program): Number of distinct prime numbers necessary to represent a natural number n > 1.
- A115591 (program): Primes p such that the multiplicative order of 2 modulo p is (p-1)/2.
- A115594 (program): Triangle read by rows: number of isomorphism classes of series-parallel matroids of rank d on n elements.
- A115598 (program): Consider all Pythagorean triples (X,X+1,Z) ordered by increasing Z; sequence gives Z-(X+1) values.
- A115599 (program): Consider all Pythagorean triples (X,X+1,Z) ordered by increasing Z; sequence gives Z-X values.
- A115601 (program): a(n) = numerator of b(n), where b(1) = 1, b(n+1) = Sum_{k=1..n} b(k)^((-1)^(n-k+1)).
- A115602 (program): a(n) = denominator of b(n), where b(1) = 1, b(n+1) = Sum_{k=1..n} b(k)^((-1)^(n-k+1)).
- A115605 (program): Expansion of -x^2*(2 + x - 2*x^2 - x^3 + 2*x^4) / ( (x-1)*(1+x)*(1 + x + x^2)*(x^2 - x + 1)*(x^2 + 4*x - 1)*(x^2 - x - 1) ).
- A115607 (program): Sum of odd divisors of n times (-1)^(n+1).
- A115618 (program): 1 + (n+6)*2^(n-1).
- A115623 (program): Irregular triangle read by rows: row n lists numbers of distinct parts of partitions of n in Mathematica order.
- A115634 (program): Expansion of (1-4*x^2)/(1-x^2).
- A115635 (program): Periodic {1,1,-5,0,1,-3,-1,0,-3,1,-1,-4}.
- A115637 (program): A divide-and-conquer sequence.
- A115638 (program): A Jacobsthal-related divide-and-conquer sequence.
- A115639 (program): First column of divide-and-conquer triangle A115636.
- A115659 (program): Permutation of natural numbers generated by 2-rowed array shown below.
- A115671 (program): Number of partitions of n into parts not congruent to 0, 2, 12, 14, 16, 18, 20, 30 (mod 32).
- A115711 (program): Squares whose digit reversal is a semiprime (A001358).
- A115716 (program): A divide-and-conquer sequence.
- A115730 (program): a(n) = a(n-3) + A001654(n-1) with a(0)=0, a(1)=0 and a(2)=1.
- A115731 (program): Permutation of natural numbers generated by 3-rowed array shown below.
- A115732 (program): n-th prime p(n) repeated (p(n)-n) times.
- A115733 (program): n-th prime minus n, p(n)-n, repeated (p(n)-n) times.
- A115754 (program): Decimal expansion of sqrt(3/2).
- A115767 (program): Integers i such that 2*i XOR 5*i = 3*i.
- A115788 (program): a(n) = floor(n*Pi) mod 2.
- A115789 (program): a(n) = (floor((n+1)*Pi) - floor(n*Pi)) mod 2.
- A115790 (program): 1 - (Floor((n+1)*Pi)-Floor(n*Pi)) mod 2.
- A115792 (program): A dihedial D1 elliptical transform on A000073.
- A115793 (program): Integers i such that i XOR 10i = 11i.
- A115794 (program): Sequence A115793 in binary.
- A115809 (program): Integers i such that 17*i = 49 X i.
- A115810 (program): Sequence A115809 in binary.
- A115836 (program): Self-describing sequence. The n-th integer of the sequence indicates how many integers of the sequence are strictly < 2n.
- A115837 (program): Self-describing sequence. The n-th integer of the sequence indicates how many integers of the sequence are strictly < 3n.
- A115845 (program): Numbers n such that there is no bit position where the binary expansions of n and 8n are both 1.
- A115846 (program): Sequence A115845 in binary.
- A115851 (program): G.f. x^2*(-1+x+x^2)/((1-x)*(2*x-1)*(x+1)*(x^2+1)).
- A115852 (program): Dihedral D3 elliptical invariant transform on A000045: a[n+1]/a[n]= Phi^4=((1+Sqrt[5])/2)^4.
- A115864 (program): Legendre_P(n,2)*4^n.
- A115865 (program): a(n) = Legendre_P(n,2)*6^n.
- A115878 (program): a(n) is the number of positive solutions of the Diophantine equation x^2 = y(y+n).
- A115880 (program): Largest positive x satisfying the Diophantine equation x^2 = y*(y+n), a(n)=0 if there are no solutions.
- A115881 (program): a(n) is the largest positive y satisfying the Diophantine equation x^2=y(y+n). a(n)=0 if there are no solutions.
- A115902 (program): Expansion of (1-8*x)^(-3/2).
- A115903 (program): Expansion of (1-12*x)^(-3/2).
- A115919 (program): Numbers k such that sigma(k) - phi(k) is a prime number.
- A115944 (program): Number of partitions of n into distinct factorials.
- A115945 (program): Numbers that cannot be written as a sum of distinct factorials.
- A115948 (program): a(n) = (2^(semiprime(n)-1)) modulo (semiprime(n)^2).
- A115951 (program): Expansion of 1/sqrt(1-4*x*y-4*x^2*y).
- A115952 (program): Expansion of (1-x+x*y)/(1-x^2*y^2) - x^2/(1-x^2*y).
- A115953 (program): Periodic {1,-1,0,0,1,-2,1,0,0,-1,1,-1}.
- A115954 (program): Inverse of number triangle A115952.
- A115955 (program): Product of A115952 and summing matrix (1/(1-x),x).
- A115956 (program): Numbers n having exactly 2 distinct prime factors, the largest of which is greater than or equal to sqrt(n) (i.e., sqrt(n)-rough numbers with exactly 2 distinct prime factors).
- A115960 (program): Numbers n having exactly 6 distinct prime factors, the largest of which is greater than or equal to sqrt(n) (i.e., sqrt(n)-rough numbers with exactly 6 distinct prime factors).
- A115962 (program): Expansion of 1/sqrt(1-4*x^2-4*x^3).
- A115963 (program): Numerator of Sum_{i=1..n} 1/prime(i)^3.
- A115964 (program): Denominator of sum_{i=1..n} 1/prime(i)^3.
- A115966 (program): Inverse permutation to sequence A094077.
- A115967 (program): Expansion of 1/(2*sqrt(1-2*x-3*x^2) - 1).
- A115969 (program): Expansion of 1/(2*sqrt(1-6*x+x^2) - 1).
- A115970 (program): Expansion of 1/(4*sqrt(1-4*x) - 3).
- A115971 (program): a(0) = 0. If a(n) = 0, then a(2^n) through a(2^(n+1)-1) are each equal to 1. If a(n) = 1, then a(m + 2^n) = a(m) for each m, 0 <= m <= 2^n -1.
- A115974 (program): Number of Feynman diagrams (vanishing and non-vanishing) of order 2n for the proper self-energy function of quantum electrodynamics (QED).
- A115975 (program): Numbers of the form p^k, where p is a prime and k is a Fibonacci number.
- A115977 (program): Expansion of elliptic modular function lambda in powers of the nome q.
- A115978 (program): Expansion of phi(-q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.
- A115979 (program): Expansion of (1 - theta_4(q)*theta_4(q^3))/2 in powers of q.
- A115990 (program): Riordan array (1/sqrt(1-2*x-3*x^2), ((1-2*x-3*x^2)/(2*(1-3*x)) - sqrt(1-2*x-3*x^2)/2).
- A115991 (program): Number triangle T(n,k) = Sum_{j=0..n} C(n-k,j-k)*C(j,n-j)*2^(n-j).
- A116000 (program): phi(n) + sigma(n) gives a semiprime (A001358).
- A116023 (program): The n-th prime plus n gives a semiprime (A001358).
- A116024 (program): The n-th prime minus n gives a semiprime (A001358).
- A116026 (program): phi(n) plus n gives a semiprime (A001358).
- A116071 (program): Triangle T, read by rows, equal to Pascal’s triangle to the matrix power of Pascal’s triangle, so that T = C^C, where C(n,k) = binomial(n,k) and T(n,k) = A000248(n-k)*C(n,k).
- A116072 (program): Central terms of triangle A116071, which equals Pascal’s triangle to the matrix power of Pascal’s triangle.
- A116073 (program): Sum of the divisors of n that are not divisible by 5.
- A116078 (program): Column 0 of triangle A116077.
- A116081 (program): Final nonzero digit of n^n.
- A116082 (program): a(n) = C(n,7) + C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).
- A116083 (program): Numbers n such that phi(sigma(n))-sigma(phi(n))=1.
- A116088 (program): Riordan array (1, x*(1+x)^2).
- A116089 (program): Riordan array (1, x*(1+x)^3).
- A116090 (program): Expansion of 1/(1-x^2*(1+x)^3).
- A116091 (program): Expansion of 1/sqrt(1+4*x+16*x^2).
- A116092 (program): Expansion of 1/sqrt(1+8*x+64*x^2).
- A116093 (program): Expansion of 1/sqrt(1+4*x+12*x^2).
- A116127 (program): Number of numbers that are congruent to {2, 4} mod 6 between prime(n) and prime(n+1) inclusive.
- A116138 (program): a(n) = 3^n * n*(n + 1).
- A116144 (program): a(n) = 4^n * n*(n+1).
- A116149 (program): a(n) = sum of n consecutive cubes after n^3.
- A116150 (program): a(n) = Sum_{j=1..n} (3^j + (-2)^j).
- A116156 (program): a(n) = 5^n * n*(n + 1).
- A116157 (program): a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) + a(n-5).
- A116164 (program): a(n) = 6^n * n*(n+1).
- A116165 (program): a(n) = 7^n * n*(n+1).
- A116166 (program): a(n) = 8^n * n*(n+1).
- A116176 (program): a(n) = 9^n * n*(n+1).
- A116178 (program): Stewart’s choral sequence: a(3n) = 0, a(3n-1) = 1, a(3n+1) = a(n).
- A116192 (program): Triangle T(n,k), 0<=k<=n : T(n,k)is smallest number such that T(n,k)>T(n-1,k-1), T(n,k)>T(n-1,k), T(n,k)and T(n-1,k-1)+T(n-1,k) have the same parity, T(0,0)=1 .
- A116201 (program): a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4); a(0)=0, a(1)=1, a(2)=1, a(3)=1.
- A116218 (program): If X_1,…,X_n is a partition of a 2n-set X into 2-blocks (or pairs) then a(n) is equal to the number of permutations f of X such that f(X_i) != X_i for all i=1,…n.
- A116362 (program): Smallest m such that A116361(m) = n.
- A116363 (program): a(n) = dot product of row n in Catalan triangle A033184 with row n in Pascal’s triangle.
- A116364 (program): Row squared sums of Catalan triangle A033184.
- A116366 (program): Triangle read by rows: even numbers as sums of two odd primes.
- A116367 (program): Sums of rows of the triangle in A116366.
- A116368 (program): Central terms of the triangle in A116366.
- A116382 (program): Riordan array (1/sqrt(1-4*x^2), (1-2*x^2*c(x^2))*(x^2*c(x^2))/(x*(1-x-x^2*c(x^2))) where c(x) is the g.f. of A000108.
- A116383 (program): Row sums of number triangle A116382.
- A116384 (program): Diagonal sums of the Riordan array A116382.
- A116385 (program): E.g.f. Bessel_I(2,2x)+2*Bessel_I(3,2x)+Bessel_I(4,2x).
- A116386 (program): Number of calendar weeks in the year n (starting at n=0 for the year 2000).
- A116387 (program): Expansion of 1/(sqrt(1-2*x-3*x^2)*(2-M(x))), where M(x) is the g.f. of the Motzkin numbers A001006.
- A116390 (program): Expansion of 1/(2*sqrt(1-4*x^2)-x-1).
- A116391 (program): Expansion of 1/((1+x)*(sqrt(1-4*x^2)-x)).
- A116394 (program): Expansion of 1/((1+x)*sqrt(1-2*x-3*x^2) - x).
- A116395 (program): Riordan array (1/sqrt(1-4*x), (1/sqrt(1-4*x)-1)/2).
- A116396 (program): Expansion of 2/((2+x)*sqrt(1-4*x)-x).
- A116399 (program): Triangle whose k-th column has e.g.f. sum{j=0..k, Bessel_I(k+j,2x)}.
- A116400 (program): E.g.f. Bessel_I(2,2x)+Bessel_I(3,2x)+Bessel_I(4,2x).
- A116404 (program): Expansion of (1-x)/((1-x)^2 - x^2*(1+x)^2).
- A116405 (program): Triangle whose k-th column has e.g.f. sum{j=0..k, (-1)^j*Bessel_I(k+j,2x)}.
- A116406 (program): Expansion of ((1 + x - 2x^2) + (1+x)*sqrt(1-4x^2))/(2(1-4x^2)).
- A116408 (program): E.g.f. exp(x)*(Bessel_I(2,2*x) - Bessel_I(3,2*x) + Bessel_I(4,2*x)).
- A116409 (program): Expansion of (1-x-2x^2+sqrt(1-2x-3x^2))/(2(1-x)(1-2x-3x^2)).
- A116410 (program): Expansion of (1-x-2x^2+sqrt(1-2x-3x^2))/(2*(1-2x-3x^2)).
- A116411 (program): Coefficient of x^n in the expansion of (1+x+x^3)^n.
- A116412 (program): Riordan array ((1+x)/(1-2x),x(1+x)/(1-2x)).
- A116413 (program): Expansion of (1+x)/(1-2x-x^2-x^3).
- A116414 (program): Riordan array (1/((1-x)(1-3x)),x/((1-x)(1-3x))).
- A116415 (program): a(n) = 5*a(n-1) - 3*a(n-2).
- A116419 (program): Reduced numerators of 2*(2^(1+n)-1)/(1+n)/(2+n).
- A116420 (program): Reduced denominators of 2(2^(1+n)-1)/(1+n)/(2+n).
- A116421 (program): a(n) = 2^(n-1)*binomial(2n-1,n-1)^2.
- A116423 (program): Binomial transform of A006053.
- A116425 (program): Decimal expansion of 2 + 2*cos(2*Pi/7).
- A116445 (program): Array read by antidiagonals: the binomial transform of the sequence (1,2,..n,0,0,0..) in row n.
- A116447 (program): a(2n) = n, a(2n+1) = n!.
- A116451 (program): Numbers having fewer prime factors than at least one smaller number.
- A116452 (program): Number of prime factors of A116451.
- A116453 (program): Third smallest number with exactly n prime factors.
- A116454 (program): Smallest m such that A116452(m) = n.
- A116466 (program): Unsigned row sums of triangle A114700.
- A116468 (program): Permutation of natural numbers generated by 2-rowed array shown below.
- A116470 (program): All distinct Fibonacci and Lucas numbers.
- A116471 (program): Values 2*(n -+ 1)^2 sorted.
- A116472 (program): a(n) = floor(exp(2*n)).
- A116477 (program): a(n) = Sum_{1<=k<=n, gcd(k,n)=1} floor(n/k).
- A116483 (program): Expansion of (1 + x) / (5*x^2 - 2*x + 1).
- A116484 (program): Expansion of (-1+3*x)/(5*x^2 + 1 - 2*x).
- A116487 (program): First digit after decimal point in decimal representation of (1+1/n)^n.
- A116494 (program): Expansion of psi(q^5)/psi(q) in powers of q where psi() is a Ramanujan theta function.
- A116508 (program): a(n) = C( C(n,2), n) for n >= 1.
- A116509 (program): Values of c in a^2 + b^2 = c^2 where b - a = 31 and gcd(a,b)=1.
- A116510 (program): a(0)=1. a(m +2^n) = a(m) - a(n), 0 <= m <= 2^n -1.
- A116512 (program): a(n) = number of positive integers each of which is <= n and is divisible by exactly one prime dividing n (but is coprime to every other prime dividing n). a(1) = 0.
- A116515 (program): a(n) = the period of the Fibonacci numbers modulo p divided by the smallest m such that p divides Fibonacci(m), where p is the n-th prime.
- A116520 (program): a(0) = 0, a(1) = 1; a(n) = max { 4*a(k) + a(n-k) | 1 <= k <= n/2 }, for n > 1.
- A116521 (program): Binomial transform of tetranacci sequence A000078.
- A116522 (program): a(0)=1, a(1)=1, a(n)=7*a(n/2) for n=2,4,6,…, a(n)=6*a((n-1)/2)+a((n+1)/2) for n=3,5,7,….
- A116523 (program): a(0)=1, a(1)=1, a(n) = 17*a(n/2) for n=2,4,6,…, a(n) = 16*a((n-1)/2) + a((n+1)/2) for n=3,5,7,….
- A116524 (program): a(0)=1, a(1)=1, a(n) = 13*a(n/2) for n=2,4,6,…, a(n) = 12*a((n-1)/2) + a((n+1)/2) for n=3,5,7,….
- A116525 (program): a(0)=1, a(1)=1, a(n) = 11*a(n/2) for even n, and a(n) = 10*a((n-1)/2) + a((n+1)/2) for odd n >= 3.
- A116526 (program): a(0)=1, a(1)=1, a(n) = 9*a(n/2) for even n >= 2, and a(n) = 8*a((n-1)/2) + a((n+1)/2) for odd n >= 3.
- A116528 (program): a(0)=0, a(1)=1, and for n>=2, a(2*n) = a(n), a(2*n+1) = 2*a(n) + a(n+1).
- A116530 (program): a(n) = 3*a(n-1), with a(1) = 20.
- A116533 (program): a(1)=1, a(2)=2, for n > 2 if a(n-1) is prime, then a(n) = 2*a(n-1), otherwise a(n) = a(n-1) - 1.
- A116541 (program): Triangular numbers for which the number of divisors is also a triangular number.
- A116543 (program): Number of terms in greedy representation of n in terms of the Lucas numbers.
- A116544 (program): Triangular numbers for which the multiplicative digital root is also a triangular number.
- A116545 (program): Sum of the largest Fibonacci exponent of prime factorizations of k, k=1..n.
- A116549 (program): a(0) = 1. a(m + 2^n) = a(n) + a(m), for 0 <= m <= 2^n - 1.
- A116551 (program): Permutation of natural numbers generated by 3-rowed array shown below.
- A116556 (program): a(n) = 2*a(n-1) + 2*a(n-2), a(0)=0, a(1)=4.
- A116558 (program): a(n) = 6*a(n-4) - a(n-8).
- A116563 (program): a(n) is the genus of the modular curve X_0(p) for p = prime(n).
- A116564 (program): Ono supersingular invariant power function.
- A116568 (program): Difference between n and the absolute value of the difference between number of nonprimes not exceeding n and number of primes not exceeding n.
- A116570 (program): a(2*n) = prime(n+1) * prime(n+2), a(2*n-1) = prime(n+1).
- A116572 (program): a(n) = floor(prime(n)/5) for n > 2, a(1) = a(2) = 1.
- A116576 (program): Number of distinct squares D(n) in the n-th iterate of the tribonacci morphism (a -> ab, b -> ac, c -> a) on the letter a.
- A116579 (program): a(1) = a(2) = 1; a(n) = floor(prime(n)/6) for n > 2.
- A116581 (program): Primes of the form k^3-k-1.
- A116588 (program): Array read by antidiagonals: T(n,k) = max(2^(n - k), 2^(k - n)).
- A116589 (program): a(n) = Sum{sqrt(n) < i <= n} i - Sum{1 <= i < sqrt(n)} i.
- A116593 (program): a(n) = b(n+2) + b(n), where b(n) = A006046(n) is the sequence defined by b(0)=0, b(1)=1, b(n) = 2*b((n-1)/2) + b((n+1)/2) for n =3,5,7,… and b(n) = 3*b(n/2) for n =2,4,6,….
- A116598 (program): Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 1 (n>=0, 0<=k<=n).
- A116599 (program): Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 2 (n>=0, 0<=k<=floor(n/2)).
- A116601 (program): a(0) = a(1) = 0; for n >= 2, a(n) = floor(sqrt(2^(n-2)-1)).
- A116603 (program): Coefficients in asymptotic expansion of sequence A052129.
- A116604 (program): Expansion of q^(-1/2) * eta(q)^3 * eta(q^4) * eta(q^12) / (eta(q^2)^2 * eta(q^3)) in powers of q.
- A116607 (program): Sum of the divisors of n which are not divisible by 9.
- A116609 (program): a(n) = 13^n modulo n.
- A116621 (program): Positive integers n such that 13^n == 1 (mod n).
- A116623 (program): a(0)=1, a(2n) = a(n)+A000079(A000523(2n)), a(2n+1) = 3*a(n) + A000079(A000523(2n+1)+1).
- A116634 (program): Number of partitions of n having exactly one part that is a multiple of 3.
- A116637 (program): G.f. satisfies: A(x) = x/series_reversion(x/G(x)) where A(x) + A(-x) = 2*G(x^2) and G(x) is the g.f. of A046646.
- A116640 (program): a(n) = A116623(A059893(n)).
- A116646 (program): Number of doubletons in all partitions of n. By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] has two doubletons, shown between parentheses).
- A116647 (program): Triangle of number of partitions that fit in an n X n box (but not in an (n-1) X (n-1) box) with Durfee square k.
- A116661 (program): Integers in both sequences A114522 and A063989.
- A116663 (program): Triangle read by rows: T(n,k) = number of partitions of n into odd parts and having exactly k parts equal to 1 (n>=0, 1<=k<=n).
- A116666 (program): Triangle, row sums = number of edges in n-dimensional hypercubes.
- A116667 (program): Greatest digit not used in n (or 10 if n is pandigital).
- A116668 (program): a(n) = (5*n^2 + n + 2)/2.
- A116669 (program): Triangle, rows tend to A001787, number of edges in n-dimensional hypercubes.
- A116670 (program): Numbers with all but one decimal digit.
- A116689 (program): Partial sums of dodecahedral numbers (A006566).
- A116690 (program): a(n) = C(n,8) + C(n,7) + C(n,6) + C(n,5) + C(n,4) + C(n,3) + C(n,2) + C(n,1).
- A116695 (program): Digit not appearing in A116670(n).
- A116697 (program): a(n) = -a(n-1) - a(n-3) + a(n-4).
- A116698 (program): Expansion of -(1-x+3*x^2+x^3) / ((x^2+x-1)*(2*x^2+1)).
- A116699 (program): Number of permutations of length n which avoid the patterns 123 and 4312.
- A116701 (program): Number of permutations of length n that avoid the patterns 132, 4321.
- A116702 (program): Number of permutations of length n which avoid the patterns 123, 3241.
- A116703 (program): Number of permutations of length n which avoid the patterns 231, 4123.
- A116706 (program): Number of permutations of length n which avoid the patterns 2134, 3421.
- A116707 (program): Number of permutations of length n which avoid the patterns 1342, 4213.
- A116709 (program): Number of permutations of length n which avoid the patterns 2341, 4213.
- A116710 (program): Number of permutations of length n which avoid the patterns 1423, 3421.
- A116711 (program): Number of permutations of length n which avoid the patterns 123, 3214, 4312.
- A116712 (program): Number of permutations of length n which avoid the patterns 231, 3214, 4312.
- A116713 (program): Number of permutations of length n which avoid the patterns 123, 2431, 4132.
- A116714 (program): Number of permutations of length n that avoid the patterns 321, 1342, 4123.
- A116715 (program): Number of permutations of length n which avoid the patterns 312, 2341, 4321.
- A116716 (program): Number of permutations of length n which avoid the patterns 321, 2341, 4123.
- A116717 (program): Number of permutations of length n which avoid the patterns 231, 1423, 3214.
- A116718 (program): Number of permutations of length n which avoid the patterns 321, 1342, 3124.
- A116720 (program): Number of permutations of length n which avoid the patterns 213, 1234, 4312.
- A116721 (program): Number of permutations of length n which avoid the patterns 123, 3142, 4312; or avoid the patterns 123, 3421, 4231.
- A116722 (program): Number of permutations of length n which avoid the patterns 312, 1324, 3421; or avoid the patterns 312, 1324, 2341, etc.
- A116723 (program): We have one bead labeled i for every i=1, 2, …; a(n) = number of necklaces that can be made using any subset of the first n beads.
- A116725 (program): Number of permutations of length n which avoid the patterns 132, 3421, 4231.
- A116726 (program): Number of permutations of length n which avoid the patterns 213, 1234, 2431.
- A116727 (program): Number of permutations of length n which avoid the patterns 321, 2134, 3412.
- A116728 (program): Number of permutations of length n which avoid the patterns 321, 1243, 2134.
- A116730 (program): Number of permutations of length n which avoid the patterns 321, 1342, 1423.
- A116731 (program): Number of permutations of length n which avoid the patterns 321, 2143, 3124; or avoid the patterns 132, 2314, 4312, etc.
- A116732 (program): a(n) = a(n-1) + a(n-2) + a(n-3) - a(n-4).
- A116733 (program): Number of permutations of length n which avoid the patterns 321, 1324, 2341.
- A116734 (program): Number of permutations of length n which avoid the patterns 231, 1432, 4123.
- A116735 (program): Number of permutations of length n which avoid the patterns 231, 1234, 4312; or avoid the patterns 312, 1234, 1432, etc.
- A116736 (program): Number of permutations of length n which avoid the patterns 1432, 2314, 2413.
- A116737 (program): Number of permutations of length n which avoid the patterns 3412, 4123, 4321.
- A116738 (program): Number of permutations of length n which avoid the patterns 3214, 4123, 4132.
- A116741 (program): Number of permutations of length n which avoid the patterns 1342, 2314, 4213.
- A116742 (program): Number of permutations of length n which avoid the patterns 1342, 2341, 4132; or avoid the patterns 2431, 3124, 4231.
- A116743 (program): Number of permutations of length n which avoid the patterns 3421, 4123, 4213.
- A116744 (program): Number of permutations of length n which avoid the patterns 1243, 1432, 4213.
- A116745 (program): Number of permutations of length n which avoid the patterns 2134, 3214, 4312.
- A116746 (program): Number of permutations of length n which avoid the patterns 1243, 4123, 4213.
- A116747 (program): Number of permutations of length n which avoid the patterns 1234, 2413, 3241.
- A116751 (program): Number of permutations of length n which avoid the patterns 2314, 2431, 3124.
- A116754 (program): Number of permutations of length n which avoid the patterns 2134, 2143, 4312.
- A116755 (program): Number of permutations of length n which avoid the patterns 1234, 2431, 3412.
- A116757 (program): Number of permutations of length n which avoid the patterns 1324, 2314, 4312.
- A116758 (program): Number of permutations of length n which avoid the patterns 1234, 1432, 2341.
- A116759 (program): Number of permutations of length n which avoid the patterns 2134, 3421, 4123.
- A116760 (program): Number of permutations of length n which avoid the patterns 2341, 3214, 4213; or avoid the patterns 1324, 2341, 3214.
- A116761 (program): Number of permutations of length n which avoid the patterns 2143, 3124, 3421.
- A116763 (program): Number of permutations of length n which avoid the patterns 2134, 3241, 3421.
- A116764 (program): Number of permutations of length n which avoid the patterns 1423, 2134, 3214.
- A116768 (program): Number of permutations of length n which avoid the patterns 1342, 3214, 4213.
- A116770 (program): Number of permutations of length n which avoid the patterns 1243, 1342, 4312.
- A116773 (program): Number of permutations of length n which avoid the patterns 1432, 2134, 4132; or avoid the patterns 3124, 4123, 4321.
- A116774 (program): Number of permutations of length n which avoid the patterns 2143, 2341, 4312; or avoid the patterns 1234, 1432, 3412.
- A116776 (program): Number of permutations of length n which avoid the patterns 2134, 3142, 3421.
- A116777 (program): Number of permutations of length n which avoid the patterns 2314, 3142, 4312.
- A116778 (program): Number of permutations of length n which avoid the patterns 2431, 3124, 3421.
- A116779 (program): Number of permutations of length n which avoid the patterns 2143, 2341, 3214.
- A116781 (program): Number of permutations of length n which avoid the patterns 1234, 1243, 3214.
- A116782 (program): Number of permutations of length n which avoid the patterns 3421, 4123, 4231; or avoid the patterns 1342, 3142, 4213.
- A116784 (program): Number of permutations of length n which avoid the patterns 2314, 3241, 4312.
- A116788 (program): Number of permutations of length n which avoid the patterns 1234, 3142, 4132.
- A116790 (program): Number of permutations of length n which avoid the patterns 1423, 1432, 3241.
- A116791 (program): Number of permutations of length n which avoid the patterns 1234, 1342, 4312.
- A116793 (program): Number of permutations of length n which avoid the patterns 1432, 2143, 3124; or avoid the patterns 1432, 2314, 3142.
- A116796 (program): Number of permutations of length n which avoid the patterns 2314, 3241, 4132.
- A116798 (program): Number of permutations of length n which avoid the patterns 1234, 1342, 1432.
- A116802 (program): Number of permutations of length n which avoid the patterns 1342, 3421, 4213.
- A116805 (program): Number of permutations of length n which avoid the patterns 2134, 3214, 4123.
- A116806 (program): Number of permutations of length n which avoid the patterns 2314, 4213, 4312.
- A116809 (program): Number of permutations of length n which avoid the patterns 1432, 2134, 2143.
- A116816 (program): Number of permutations of length n which avoid the patterns 2314, 3124, 4312.
- A116817 (program): Number of permutations of length n which avoid the patterns 2341, 3241, 4132.
- A116819 (program): Number of permutations of length n which avoid the patterns 2431, 4123, 4231.
- A116820 (program): Number of permutations of length n which avoid the patterns 2341, 3241, 4213.
- A116823 (program): Number of permutations of length n which avoid the patterns 1342, 3142, 4312; or avoid the patterns 3124, 3412, 3421.
- A116826 (program): Number of permutations of length n which avoid the patterns 2143, 2431, 3124.
- A116837 (program): Number of permutations of length n which avoid the patterns 3421, 4123, 4312; or avoid the patterns 2341, 3142, 3214.
- A116844 (program): Number of permutations of length n which avoid the patterns 231, 12345.
- A116845 (program): Number of permutations of length n which avoid the patterns 231, 12534.
- A116847 (program): Number of permutations of length n which avoid the patterns 123, 51432.
- A116848 (program): Number of permutations of length n which avoid the patterns 231, 51234.
- A116849 (program): Number of permutations of length n which avoid the patterns 213, 51432.
- A116850 (program): Number of permutations of length n which avoid the patterns 231, 12354.
- A116851 (program): Number of permutations of length n which avoid the patterns 321, 31245.
- A116852 (program): Number of partitions of n-th semiprime into 2 squares.
- A116853 (program): Difference triangle of factorial numbers read by upward diagonals.
- A116854 (program): First differences of the rows in the triangle of A116853, starting with 0.
- A116855 (program): Triangle read by rows, constructed from binomial transforms of prefixes of A000255 (see Comments for precise definition).
- A116862 (program): Row sums of triangle A116868 (called Y(1,3)).
- A116866 (program): Generalized Catalan triangle of Riordan type, called C(1,3).
- A116867 (program): Convolution of generalized Catalan sequence A064063 (named C(3;n)).
- A116871 (program): Sixth column of triangle A067323.
- A116873 (program): Generalized Catalan numbers C(2,3;n)=C(3,2;n).
- A116879 (program): Row sums of triangle A116872.
- A116881 (program): Row sums of triangle A116880 (generalized Catalan C(1,2)).
- A116882 (program): A number n is included if (highest odd divisor of n)^2 <= n.
- A116883 (program): A number k is included iff (highest odd divisor of k)^2 >= k.
- A116891 (program): a(n) = gcd(n! + 1, n^n + 1).
- A116895 (program): Least prime factor of n^n-1.
- A116905 (program): Number of partitions of n-th 3-almost prime into 2 squares.
- A116906 (program): Sum of squares of divisors of n!.
- A116913 (program): Inverse Moebius transform of pentagonal numbers.
- A116914 (program): Number of UUDD’s, where U=(1,1) and D=(1,-1), in all hill-free Dyck paths of semilength n (a hill in a Dyck path is a peak at level 1).
- A116915 (program): Expansion of f(-x, -x^4)^2 / f(-x, -x^2) in powers of x where f(, ) is Ramanujan’s general theta function.
- A116916 (program): Expansion of q^(-1/8) * (eta(q)^3 + 3 * eta(q^9)^3) in powers of q^3.
- A116917 (program): a(1)=a(2)=1. a(n) = A006530(a(n-1)) + A006530(a(n-2)).
- A116919 (program): a(0)=1. a(n) = A006530(a(n-1)) + n.
- A116920 (program): a(0)=1. a(n) = A020639(a(n-1)) + n.
- A116921 (program): a(n) = largest integer <= n/2 which is coprime to n.
- A116922 (program): a(n) = smallest integer >= n/2 which is coprime to n.
- A116924 (program): Triangle T(n,k) = B(k,n) - B(k-1,n) where B(n,m) = Sum_{i=0..n} binomial(m,i) (3*i+1).
- A116928 (program): Number of 1’s in all self-conjugate partitions of n.
- A116937 (program): Expansion of Pi^2 in base 2.
- A116938 (program): Expansion of e^2 in base 2.
- A116939 (program): Lexicographically earliest sequence such that each i occurs exactly i+1 times and succeeding terms differ exactly by -1 or +1.
- A116940 (program): Greatest m such that A116939(m) = n.
- A116941 (program): Permutation of the natural numbers in conjunction with A116939 and A003056.
- A116942 (program): Permutation of the natural numbers in conjunction with A116939 and A003056.
- A116943 (program): Number of 4s digits plus non-final 3s digits 3 base 5 expansion of 2^n.
- A116948 (program): Riordan array ((1+2x^2)/(1-x^3),x).
- A116949 (program): Riordan array ((1-x^3)/(1+2x^2),x).
- A116952 (program): a(n) = 3*a(n-1) + 5 with a(0) = 1.
- A116953 (program): a(n) = Floor[1/2((1-2/Sqrt[3])^n+(1-2/Sqrt[3])^n)]
- A116954 (program): Numbers n such that 3*n^3 + 1 is prime.
- A116955 (program): a(n+1) = a(n) + (if a(n) is odd then (next odd square) else (next even square)), a(0) = 1.
- A116956 (program): Number of functions f:{1,2,…,n}->{1,2,…,n} with odd cycles only.
- A116958 (program): Numbers k such that 2*k + 5 and 2*k + 7 are twin primes.
- A116963 (program): Inverse Moebius transform of tetrahedral numbers.
- A116966 (program): a(n) = n + {1,2,0,1} according as n == {0,1,2,3} mod 4.
- A116969 (program): If n mod 2 = 0 then 3*2^(n-1)+n-1 else 3*2^(n-1)+n.
- A116970 (program): a(n) = (3^n - 7)/2.
- A116971 (program): a(n) = (35*2^((2*(3*n+2) + 2)/3) - 2*(3*n+2) - 46)/9.
- A116972 (program): a(n) = 11*3^n - 2*n - 10.
- A116973 (program): If n mod 2 = 0 then (3^(n+3)-19)/8 else (3^(n+3)-1)/8.
- A116974 (program): Numbers n for which the sum of the proper divisors equals the product of the proper divisors.
- A116975 (program): Number of compositions of n into parts of sizes == 1 mod 5 or 4 mod 5.
- A116982 (program): Distance between prime neighbors of 4n.
- A116995 (program): Pentagonal numbers with prime indices.
- A116996 (program): Partial sums of A116966.
- A116998 (program): Numbers having no fewer distinct prime factors than any predecessor; a(1) = 1.
- A117000 (program): a(n) = Sum_{d|n} Jacobi(2,d)*d.
- A117002 (program): a(n) = sigma(n) + 3*A079667(n).
- A117003 (program): a(n) = sigma(n) + A079667(n).
- A117004 (program): a(n) = sigma(n) - A079667(n).
- A117013 (program): Decimal expansion of (sine of 1 radian)^2.
- A117015 (program): Decimal expansion of (sine of 1 radian)^4.
- A117030 (program): a(1) = a(2) = 1; a(n) = a(n-1)*a(n-2) - a(n-3) - a(n-4) - … - a(1) for n>2.
- A117033 (program): Decimal expansion of (cos 1)^2.
- A117035 (program): Decimal expansion of (cos 1)^4.
- A117047 (program): Primes of the form 60*n+11.
- A117049 (program): Primes of the form 22*(n^2)+1.
- A117054 (program): Number of ordered ways of writing n = i + j, where i is a prime and j is of the form k*(k+1), k > 0.
- A117060 (program): Mersenne numbers for which the product of the digits is not zero.
- A117061 (program): Numbers n such that a(n) = (s(n-1))^2+n, with a(1) = 1.
- A117062 (program): Hexagonal numbers for which the sum of the digits is also a hexagonal number.
- A117066 (program): Partial sums of cupolar numbers (1/3)*(n+1)*(5*n^2+7*n+3) (A096000).
- A117067 (program): Decimal value of binary number whose n-th 1 is followed by F(n) 0’s.
- A117077 (program): Define binary strings S(0)=0, S(1)=1, S(n) = S(n-2)S(n-1); a(n) = S(n) converted to decimal.
- A117079 (program): a(1) = 1; a(n) = 2*a(n-1) + (number of digits in a(n-1)).
- A117080 (program): a(n) = 2a(n-1)+a(n-3)+1 with a(1)=1, a(2)=3, a(3)=8.
- A117081 (program): a(n) = 36*n^2 - 810*n + 2753, producing the conjectured record number of 45 primes in a contiguous range of n for quadratic polynomials, i.e., abs(a(n)) is prime for 0 <= n < 44.
- A117085 (program): Decimal value of binary number whose n-th 1 is followed by L(n) 0’s.
- A117088 (program): a(n) = (11*5^n - 7)/4.
- A117110 (program): The (1,1)-entry of the vector v[n]=Mv[n-1], where M is the 3 x 3 matrix [[0,-1/r,r],[ -1/r,-2/r,1],[r,1,2+2/r]], r being the golden ratio and v[0] is the column matrix [0,1,1].
- A117119 (program): Number of partitions of 2*n into two odd prime powers.
- A117120 (program): a(1)=1. a(n) is smallest positive integer not occurring earlier in the sequence where a(n) is congruent to -1 (mod a(n-1)).
- A117123 (program): n minus the number of 0’s in binary expansion of n.
- A117124 (program): Numbers that when multiplied by 37 produce a palindrome number.
- A117131 (program): Remainder when n^n is divided by the n-th prime number.
- A117142 (program): Number of partitions of n in which any two parts differ by at most 2.
- A117143 (program): Number of partitions of n in which any two parts differ by at most 3.
- A117149 (program): Trajectory of 4 under map k -> A094077(k).
- A117150 (program): Retrograde trajectory of 4 under map k -> A094077(k).
- A117152 (program): Sum of product of Fibonacci and triangular numbers.
- A117157 (program): a(1)=a(2)=1; a(n) = a(n-1)*a(n-2) + a(n-3) + a(n-4) + … + a(1) for n>2.
- A117178 (program): Riordan array (c(x^2)/sqrt(1-4*x^2), x*c(x^2)), c(x) the g.f. of A000108.
- A117179 (program): Riordan array ((1-x^2)/(1+x^2)^2,x/(1+x^2)).
- A117180 (program): Lowest prime-power dividing the n-th nonsquarefree positive integer.
- A117181 (program): Highest prime-power dividing the n-th nonsquarefree positive integer.
- A117183 (program): a(n) = smallest prime dividing n-th nonsquarefree positive integer.
- A117184 (program): Riordan array ((1+x)c(x^2)/sqrt(1-4x^2),xc(x^2)), c(x) the g.f. of A000108.
- A117186 (program): Expansion of (1+x)c(x^2)/((1-xc(x^2))*sqrt(1-4x^2)), c(x) the g.f. of A000108.
- A117187 (program): Expansion of (1+x)c(x^2)/((1-x^2*c(x^2))sqrt(1-4x^2)), c(x) the g.f. of A000108.
- A117188 (program): Expansion of (1-x^2)/(1+x^2+x^4).
- A117189 (program): Binomial transform of the tribonacci sequence A000073 (shifted left twice).
- A117197 (program): a(n) = (n^3 - 1)^3.
- A117198 (program): Generalized Riordan array (1,x)+(x,x^2)+(x^2,x^3).
- A117199 (program): Expansion of 1/(1-x^2) + x/(1-x^3) + x^2/(1-x^4).
- A117202 (program): Binomial transform of n*F(n).
- A117203 (program): Odd squarefree positive integers k such that (k-1)/2 is also squarefree.
- A117204 (program): Squarefree positive integers k such that 2*k+1 is also squarefree.
- A117205 (program): Odd squarefree positive integers k such that (k+1)/2 is also squarefree.
- A117206 (program): Squarefree positive integers k such that 2*k-1 is also squarefree.
- A117207 (program): Number triangle read by rows: T(n,k)=sum{j=0..n-k, C(n+j,j+k)C(n-j,k)}.
- A117208 (program): G.f. A(x) satisfies (1-x) = product_{n>=1} A(x^n).
- A117209 (program): G.f. A(x) satisfies 1/(1-x) = product_{n>=1} A(x^n).
- A117210 (program): G.f. A(x) satisfies (1+x) = product_{n>=1} A(x^n).
- A117211 (program): G.f. A(x) satisfies 1/(1+x) = product_{n>=1} A(x^n).
- A117212 (program): Sum_{d|n} a(d)/d = (-1)^(n-1)/n for n>=1; equals the logarithmic g.f. of A117210.
- A117213 (program): a(n) = smallest term of sequence A002110 divisible by n-th squarefree positive integer.
- A117214 (program): a(n) = (A117213(n))/(n-th squarefree positive integer).
- A117216 (program): Number of points in the standard root system version of the D_4 lattice having L_infinity norm n.
- A117218 (program): Squares divisible by their number of digits.
- A117220 (program): Number of partitions of 3-smooth numbers into parts not greater than 3.
- A117229 (program): Number of decimal digits of n in {0,1,4,8,9}
- A117230 (program): Start with 1 and repeatedly reverse the digits and add 1 to get the next term.
- A117245 (program): Partial sums of A115975.
- A117248 (program): Number of down steps at start of segment n of A079051.
- A117251 (program): Column 0 of triangle A117250.
- A117253 (program): Column 0 of triangle A117252.
- A117255 (program): Column 0 of triangle A117254.
- A117257 (program): Column 0 of triangle A117256.
- A117259 (program): Column 0 of triangle A117258.
- A117260 (program): Triangle T, read by rows, where matrix inverse T^-1 has -2^n in the secondary diagonal: T^-1 = -2^n, with all 1’s in the main diagonal and zeros elsewhere.
- A117261 (program): Row sums of triangle A117260.
- A117262 (program): Triangle T, read by rows, where matrix inverse T^-1 has -3^n in the secondary diagonal: T^-1 = -3^n, with all 1’s in the main diagonal and zeros elsewhere.
- A117263 (program): Row sums of triangle A117262; also, self-convolution of A117264.
- A117266 (program): Row sums of triangle A117265.
- A117267 (program): Difference row triangle of A117189.
- A117268 (program): Triangle, binomial transform of the tribonacci sequence.
- A117274 (program): Triangle read by rows: T(n,k) is the number of partitions of n with no even part repeated and having k 1’s (n>=0, 0<=k<=n).
- A117275 (program): Number of partitions of n with no even parts repeated and with no 1’s.
- A117276 (program): Number of 1’s in all partitions of n with no even parts repeated.
- A117277 (program): Number of partitions of n whose consecutive parts differ by 3.
- A117284 (program): Numbers k for which the cototient k-phi(k) is a triangular number.
- A117286 (program): Numbers k for which the cototient k-phi(k) is a hexagonal number.
- A117290 (program): Numbers k for which the cototient k - phi(k) is a Fibonacci number.
- A117291 (program): a(n) = phi(n)^(n-phi(n))
- A117292 (program): a(n) = (n-phi(n))^phi(n).
- A117295 (program): a(n) = phi(n)*(n - phi(n)).
- A117298 (program): Number of partitions of n with unique smallest part and unique largest part.
- A117302 (program): Number of cases in which the first player gets killed in a Russian roulette game when 7 players use a gun with n chambers and the number of the bullets can be from 1 to n. In the game they do not rotate the cylinder after the game starts.
- A117303 (program): Self-inverse permutation of the natural numbers based on the bijection (2*x-1)*2^(y-1) <–> (2*y-1)*2^(x-1).
- A117305 (program): Triangular numbers for which the sum of the digits is a pentagonal number.
- A117309 (program): Triangular numbers for which the sum of the digits is a hexagonal number.
- A117316 (program): Riordan array ((1-x)/(1-x-2x^2),x(1-x)/(1-x-2x^2)).
- A117317 (program): Triangle related to partitions of n.
- A117322 (program): a(n) = prime(n) modulo semiprime(n).
- A117323 (program): Semiprime(n) modulo prime(n).
- A117336 (program): Column 1 of triangle A117335.
- A117338 (program): Row sums of triangle A117335.
- A117339 (program): a(n)=a(n-1)+a(n-2); if a(n) is not prime divide a(n) by its largest prime factor.
- A117352 (program): Riordan array (1/(1-2x), x(1-2x)/(1-x)).
- A117353 (program): Expansion of (1-x)/(1-3x+x^2+4x^3-4x^4).
- A117354 (program): Riordan array (1-x+sqrt(1-6x+x^2))/2, (1+x-sqrt(1-6x+x^2))/4).
- A117360 (program): Numbers m such that m and 2*m+1 have the same number of prime factors.
- A117362 (program): Riordan array (1-2x,x(1-x)).
- A117363 (program): Expansion of (1-2x)/(1-x^2+x^3).
- A117364 (program): a(n) = largest prime less than the largest prime dividing n (or 1 if there is no such prime).
- A117365 (program): a(n) = largest prime less than the smallest prime dividing n (or 1 if there is no such prime).
- A117366 (program): a(n) = smallest prime greater than the largest prime dividing n.
- A117367 (program): a(n) = smallest prime greater than the smallest prime dividing n.
- A117368 (program): a(n) = largest prime less than the smallest prime dividing (2n-1).
- A117370 (program): Number of primes between smallest prime divisor of n and largest prime divisor of n.
- A117372 (program): Riordan array (1-3x,x(1-x)).
- A117373 (program): Expansion of (1 - 3x)/(1 - x + x^2).
- A117374 (program): Expansion of (1-3x)/(1-x^2+x^3).
- A117375 (program): Riordan array (1/(1-3x*c(x)),xc(x)), c(x) the g.f. of A000108.
- A117377 (program): Riordan array (1-4x,x(1-x)).
- A117378 (program): Expansion of (1-4*x)/(1-x+x^2).
- A117379 (program): Expansion of (1-4x)/(1-x^2+x^3).
- A117380 (program): Riordan array (1/(1-4*x*c(x)),xc(x)), c(x) the g.f. of A000108.
- A117384 (program): Positive integers, each occurring twice in the sequence, such that a(n) = a(k) when n+k = 4*a(n), starting with a(1)=1 and filling the next vacant position with the smallest unused number.
- A117385 (program): Positive integers, each occurring twice in the sequence, such that a(n) = a(k) when n+k = 5*a(n), starting with a(1)=1 and filling the next vacant position with the smallest unused number.
- A117389 (program): A skew Delannoy number triangle read by rows.
- A117394 (program): Product of the first F(n) primes, where F(n) is the n-th Fibonacci number.
- A117396 (program): Triangle, read by rows, defined by: T(n,k) = (k+1)*T(n,k+1) - Sum_{j=1..n-k-1} T(j,0)*T(n,j+k+1) for n>k with T(n,n)=1 for n>=0.
- A117397 (program): Column 3 of triangle A117396.
- A117399 (program): Column 1 (divided by 2) of triangle A117398, which is the matrix log of A117396.
- A117401 (program): Triangle T(n,k) = 2^(k*(n-k)), read by rows.
- A117402 (program): Row sums of triangle A117401: a(n) = Sum_{k=0..n} 2^((n-k)*k) for n>=0.
- A117403 (program): a(n) = Sum_{k=0..floor(n/2)} 2^((n-2*k)*k) for n>=0.
- A117404 (program): Triangular numbers for which the sum of the digits is a square.
- A117407 (program): a(n) = j if n is T(j), else a(n) = k if n is U(k), where T is a Beatty sequence based on (sqrt(5)+5)/2 (A054770) and U is its complement (A063732).
- A117409 (program): Number of partitions of n into odd parts in which the largest part occurs only once.
- A117410 (program): Expansion of q^(-5/24) * eta(q^2)^3 / eta(q) in powers of q.
- A117411 (program): Skew triangle associated to the Euler numbers.
- A117412 (program): Sum of the interior angles of an n-sided polygon, in gradians.
- A117413 (program): Expansion of (1-x^2)/(1-2*x^2+4*x^3+x^4).
- A117415 (program): E.g.f. (x*tan(x)-x^2)/8 (even powers only).
- A117434 (program): Expansion of c(x*y(1+x)), c(x) the g.f. of A000108.
- A117435 (program): Triangle related to exp(x)*cos(2*x).
- A117436 (program): Triangle related to exp(x)*sec(2*x).
- A117437 (program): Expansion of e.g.f.: exp(x)*sec(2*x).
- A117438 (program): Triangle T(n, k) = binomial(2*n-k, k)*(-4)^(n-k), read by rows.
- A117439 (program): Expansion of (1-x^2)/(1 -4*x -2*x^2 +x^4).
- A117440 (program): A cyclically signed version of Pascal’s triangle.
- A117441 (program): Periodic with repeating part {1,1,0,1,-1,0,-1,-1,0,-1,1,0}.
- A117442 (program): Number triangle read by rows, related to exp(x)/(cos(x) + sin(x)).
- A117443 (program): Expansion of e.g.f.: exp(x)/(cos(x) + sin(x)).
- A117444 (program): Period 5: Repeat [0, 1, 2, -2, -1].
- A117445 (program): Periodic {0,-1,1,4,-1,4,-4,-4,1,1,-4,-4,4,-1,4,1,-1} (period 17).
- A117446 (program): Triangle T(n, k) = binomial(L(k/3), n-k) where L(j/p) is the Legendre symbol of j and p, read by rows.
- A117447 (program): Expansion of (1 + 2*x + 3*x^2 + x^3)/(1 + x - x^3 - x^4).
- A117448 (program): Diagonal sums of number triangle A117446.
- A117450 (program): Expansion of (1-x+x^2+x^5)/((1-x)^2*(1-x^5)).
- A117451 (program): Expansion of (1-x+x^2+x^5)/((1-x)*(1-x^5)).
- A117452 (program): Periodic {2, -1, 1, 0, 0} - 0^n.
- A117464 (program): Triangular numbers for which the product of the digits is a square.
- A117472 (program): Values of c in a^2 + b^2 = c^2, where b - a = 17 and gcd(a,b,c)=1.
- A117473 (program): The values of ‘a’ in a^2 + b^2 = c^2, where b - a = 17 and gcd(a, b, c) = 1.
- A117474 (program): The values of ‘a’ in a^2 + b^2 = c^2 where b - a = 7 and gcd(a,b,c)=1.
- A117475 (program): The values of c in a^2 + b^2 = c^2 where b - a = 23 and gcd(a,b,c) = 1.
- A117476 (program): The values of a in a^2 + b^2 = c^2 where b - a = 23 and gcd(a,b,c)=1.
- A117479 (program): Number of zeros in the maximal Fibonacci bit-representation of n (A104326).
- A117481 (program): a(n) = n*(n-1)*(n-2)*(n-3)*…*(n-k) such that (n-k) is the largest prime smaller than n.
- A117485 (program): Expansion of x^9/((1-x)*(1-x^2)*(1-x^3))^2.
- A117490 (program): Number of primes between n and n^2 (with n and n^2 excluded).
- A117494 (program): a(n) is the number of m’s, 1 <= m <= n, where gcd(m,n) is prime.
- A117495 (program): Product of a prime number p and the number of primes smaller than p.
- A117501 (program): Triangle generated from an array of generalized Fibonacci-like terms.
- A117502 (program): Triangle, row sums = A001595.
- A117507 (program): Numerators of partial sums of the Brun series divided by 4.
- A117508 (program): Denominators of partial sums of the Brun series divided by 5.
- A117512 (program): Triangular numbers for which the sum of the digits is a prime number.
- A117513 (program): Number of ways of arranging 2*n tokens in a row, with 2 copies of each token from 1 through n, such that between every pair of tokens labeled i (i = 1..n-1) there is exactly one taken labeled i+1.
- A117516 (program): Last entry (and high point) in segment n of A079051.
- A117520 (program): Triangular numbers for which the digital root is also a triangular number.
- A117521 (program): Start with 1 and repeatedly reverse the digits and add 2 to get the next term.
- A117523 (program): Triangular numbers for which the sum of the digits is an octagonal number.
- A117530 (program): Triangle read by rows: T(n,k) = k^2 - k + prime(n), 1<=k<=n.
- A117535 (program): Number of ways of writing n as a sum of powers of 3, each power being used at most 4 times.
- A117548 (program): Values of n for which there exist d(1),…,d(n), each in {0,1,2} and an r in {1,2} such that Sum[d(i)d(i+k),i=1,n-k]=r (mod 3) for all k=0,…,n-1. (Such a sequence is called a very(3,r) sequence. See the link.).
- A117552 (program): Largest partial sum of the increasingly ordered divisors of n, not exceeding n.
- A117553 (program): When adding some positive divisors of n in order from lowest divisor to highest divisor, a(n) is lowest sum achievable which is >= n.
- A117560 (program): a(n) = n*(n^2 - 1)/2 - 1.
- A117561 (program): Floor(n*(n^3-n-3)/(2*(n-1))).
- A117567 (program): Riordan array ((1+x^2)/(1-x^3),x).
- A117568 (program): Riordan array ((1-x^3)/(1+x^2),x).
- A117569 (program): Expansion of (1+x+x^2)/(1+x^2).
- A117571 (program): Expansion of (1+2*x^2)/((1-x)*(1-x^3)).
- A117572 (program): Expansion of (1+2x^2)/((1-x^2)(1-x^3)).
- A117573 (program): Expansion of (1+2x^2)/((1-x)(1-x^2)(1-x^3)).
- A117575 (program): Expansion of (1-x^3)/((1-x)*(1+2*x^2)).
- A117576 (program): Expansion of (1-x^3)/((1-x^2)(1+2x^2)).
- A117584 (program): Generalized Pellian triangle.
- A117585 (program): a(n) = 2*a(n-1) + a(n-2) + n.
- A117589 (program): Periodic with period 8: repeat 0, 1, 3, 7, 15, 14, 12, 8.
- A117590 (program): a(n) = ceiling(x(n)), where x(n) = 3*x(n-1)/2 and x(1) = 1.
- A117591 (program): a(n) = 2^n + Fibonacci(n).
- A117592 (program): a(n) = a(3n) = a(3n+1) = a(3n+2)/2 with a(0)=1.
- A117596 (program): Start with x=6/5; repeatedly apply the map x -> x*ceiling(x); sequence gives numerators of the resulting sequence of fractions.
- A117605 (program): Decimal expansion of the real solution to equation x^3 + 3*x = 2.
- A117609 (program): Number of lattice points inside the ball x^2 + y^2 + z^2 <= n.
- A117611 (program): Legendre-binomial transform of 10^n for p=3.
- A117614 (program): a(0)=1, a(n)=4a(n-1)+2 for n odd, a(n)=4a(n-1) for n even.
- A117615 (program): a(0)=0, a(1)=1, a(n)=4a(n-1)+2 for n =3,5,7,…, a(n)=4a(n-1) for n =2,4,6,….
- A117616 (program): a(0)=0, a(n)=4a(n-1)+2 for n odd, a(n)=4a(n-1) for n even.
- A117617 (program): a(n) = 5*a(n-1) + 3 with a(0) = 1.
- A117619 (program): a(n) = n^2 + 7.
- A117625 (program): Maximum number of regions defined by n zigzag-lines in the plane when a zigzag-line is defined as consisting of two parallel infinite half-lines joined by a straight line segment.
- A117627 (program): Let f(n) = minimum of average number of comparisons needed for any sorting method for n elements and let g(n) = n!*f(n). Sequence gives a lower bound on g(n).
- A117630 (program): Complement of A056576.
- A117634 (program): a(0)=0. a(n) = a(n-1) + 1 + (number of positive integers which are <= n and are missing from {a(0),a(1),a(2),…a(n-1)}).
- A117640 (program): Concatenation of first n numbers in base 4.
- A117641 (program): Number of 3-Motzkin paths of length n with no level steps at height 0.
- A117642 (program): a(n) = 3*n^3.
- A117643 (program): a(n) = n*(a(n-1)-1) starting with a(0)=3.
- A117644 (program): Number of distinct pairs a < b with nonzero decimal digits such that a + b = 10^n.
- A117647 (program): a(2n) = A014445(n), a(2n+1) = A015448(n+1).
- A117658 (program): Number of solutions to x^(k+1) = x^k mod n for some k >= 1.
- A117662 (program): n*(n-1)*(n-2)*(n+3)/12.
- A117663 (program): Heptagonal numbers for which the digital root is also a heptagonal number.
- A117664 (program): Denominator of the sum of all elements in the n X n Hilbert matrix M(i,j) = 1/(i+j-1), where i,j = 1..n.
- A117665 (program): n times the n-th n-gonal number.
- A117666 (program): Expansion of (1-3*x+x^2)*(1-x-x^2)/((1+x+x^2)*(1-x+x^2)*(1-x)^2).
- A117667 (program): a(n) = n^n-n^(n-1)-n^(n-2)-n^(n-3)-…-n^3-n^2-n.
- A117670 (program): Triangle read by rows: partial sums of the Pascal triangle minus 1.
- A117671 (program): a(n) = binomial(3*n+1, n+1).
- A117672 (program): Numbers n such that |cos(n)*cos(n+2)| < (cos(n+1))^2.
- A117673 (program): a(n) is the least k such that k*2*prime(n) + 1 is prime.
- A117676 (program): Squares for which the digital root is also a square.
- A117677 (program): a(n) = number of divisors of n^2 (excluding 1 and n^2).
- A117678 (program): Squares for which the multiplicative digital root is also a square.
- A117681 (program): Floor of exp(n^2).
- A117686 (program): Squares for which the product of the digits are cubes.
- A117689 (program): Cubes for which the product of the digits is a square.
- A117691 (program): Expansion of -(x^7+x^6+x^5-2*x^3-3*x^2-3*x-4) / ((x-1)^2*(x+1)^2*(x^2+1)^2).
- A117694 (program): (n^n + n)/2.
- A117704 (program): Least refined sequence that can be grouped to sum to either natural numbers or odd numbers.
- A117714 (program): a(n) = (A034962(n) - A152470(n))/2.
- A117715 (program): Triangle T(n,m) containing the value of the Fibonacci polynomial F(n,x) at x=m.
- A117717 (program): Maximal number of regions obtained by a straight line drawing of the complete bipartite graph K_{n,n}.
- A117718 (program): Number of heptagonal numbers with n digits.
- A117719 (program): a(2n) = A001653(n) (Numbers n such that 2*n^2 - 1 is a square), a(2n+1) = A038725(n+1).
- A117722 (program): a(n) = A000045(A003622(n)).
- A117727 (program): Partial sums of A051109.
- A117731 (program): Numerator of the fraction n*Sum_{k=1..n} 1/(n+k).
- A117733 (program): Sum of the n-th primorial and the n-th compositorial number.
- A117734 (program): Absolute difference between the n-th primorial and the n-th compositorial number.
- A117735 (program): a(n) = n! - primorial(n).
- A117736 (program): factorial(n) - A049614(n).
- A117748 (program): Triangular numbers divisible by 3.
- A117756 (program): Squares for which the reversed sum of the digits is also a square.
- A117760 (program): Expansion of 1/(1 - x - x^3 - x^5 - x^7).
- A117761 (program): a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-7) for n >= 7.
- A117762 (program): a(1)=6; for n>1, a(n) = prime(n)*(prime(n)^2-1)/2.
- A117767 (program): a(n) is the differences between the smallest square greater than prime(n) and the largest square less than prime(n), where prime(n) = A000040(n) is the n-th prime number.
- A117768 (program): Number of Lucas numbers with n digits.
- A117791 (program): Expansion of 1/(1 - x - x^2 + x^4 - x^6).
- A117792 (program): First entry of the vector (M^n)w, where M is the 6x6 matrix [[0, 1, 0, 0, 0, 0, ], [0, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1], [1, 0, -1, 0, 1, 1]] and w is the column vector [0, 1, 1, 2, 3, 5].
- A117793 (program): Pentagonal numbers divisible by 5.
- A117794 (program): Hexagonal numbers divisible by 6.
- A117795 (program): Heptagonal numbers divisible by 7.
- A117796 (program): Enneagonal numbers divisible by 9.
- A117797 (program): Decagonal numbers divisible by 10.
- A117798 (program): Icosagonal numbers divisible by 20.
- A117800 (program): Start with 1 and repeatedly reverse the digits and add 5 to get the next term.
- A117802 (program): Numbers with an “a” in Dutch.
- A117804 (program): Natural position of n in the string 12345678910111213….
- A117806 (program): n appears {a(1)+a(2)+…+a(n-1)} times (with a(1)=1).
- A117807 (program): Start with 1015 and repeatedly reverse the digits and add 4 to get the next term.
- A117812 (program): a(n) = n^(2*n) - 1.
- A117813 (program): Consider 1-D random walk with jumps up to the third neighbor, i.e., set of possible jumps is {-3,-2,-1,+1,+2,+3}. Sequence gives number of paths of length n ending at origin.
- A117814 (program): a(n) = 1 if at least one of decimal digits of n is a prime, otherwise a(n)=0.
- A117818 (program): a(n) = n if n is 1 or a prime, otherwise a(n) = n divided by the least prime factor of n (A032742(n)).
- A117824 (program): a(0) = 0, a(1) = 1; for n >= 2, a(n) = a(n-1) + a(n-2) - (n-1) if that number is positive and not already in the sequence, otherwise a(n) = a(n-1) + a(n-2) + (n-1).
- A117826 (program): First four terms of the sequence are doubled, then those numbers are tripled and then those numbers are quadrupled, etc.
- A117828 (program): Start with 1 and repeatedly reverse the decimal digits and add 4 to get the next term.
- A117829 (program): Start with 3 and repeatedly reverse the digits and add 4 to get the next term.
- A117830 (program): Let S_m be the infinite sequence formed by starting with m and repeatedly reversing the digits and adding 4 to get the next term. For all m < 1015, S_m enters the cycle of length 54 whose terms are shown here.
- A117841 (program): Start with 1 and repeatedly reverse the digits and add 10 to get the next term.
- A117842 (program): Partial sum of smallest prime >= n (A007918).
- A117849 (program): a(n) =(A001359[n]^2-1)/2
- A117852 (program): Mirror image of A098473 formatted as a triangular array.
- A117854 (program): Let p(n) be the n-th-prime. Sequence gives primes of the form | p(n)*p(n+2) - p(n+1)*p(n+3)| +1.
- A117855 (program): Number of nonzero palindromes of length n (in base 3).
- A117856 (program): Number of palindromes of length n (in base 4).
- A117857 (program): Number of palindromes of length n (in base 5).
- A117858 (program): Number of palindromes of length n (in base 6).
- A117859 (program): Number of palindromes of length n (in base 7).
- A117860 (program): Number of palindromes of length n (in base 8).
- A117861 (program): Number of palindromes of length n (in base 9).
- A117862 (program): Number of palindromes (in base 3) below 3^n.
- A117863 (program): Number of palindromes (in base 4) below 4^n.
- A117864 (program): Number of palindromes (in base 5) below 5^n.
- A117865 (program): Number of palindromes (in base 6) below 6^n.
- A117866 (program): Number of palindromes (in base 7) below 7^n.
- A117867 (program): Number of palindromes (in base 8) below 8^n.
- A117868 (program): Number of palindromes (in base 9) below 9^n.
- A117869 (program): Partial sums of floor(e^n).
- A117880 (program): a(1) = 4; a(n) is smallest semiprime > 2*a(n-1).
- A117886 (program): Expansion of q^(-2/3)eta(q)eta(q^10)^2/eta(q^5) in powers of q.
- A117887 (program): Number of labeled trees on <= n nodes.
- A117890 (program): Numbers n such that number of non-leading 0’s in binary representation of n divides n.
- A117892 (program): Add up the positive integers which are coprime to n in order (starting at 1). a(n) is the largest such partial sum that is <= n.
- A117893 (program): Add up the positive integers that are coprime to n in order (starting at 1). a(n) is the smallest such partial sum that is >= n.
- A117894 (program): Triangle, row sums = Pell numbers, A000129.
- A117895 (program): Triangle T(n, k) = (k-n)*A000129(k+1) + (3*n-3*k+1)*A000129(k) with T(n,0) = 1, for 0 <= k <= n-1, read by rows.
- A117897 (program): Number of labeled trees on prime numbers of nodes through n-th prime.
- A117898 (program): Number triangle 2^abs(L(C(n,2)/3) - L(C(k,2)/3))*[k<=n] where L(j/p) is the Legendre symbol of j and p.
- A117899 (program): Expansion of (1 + 2*x + 5*x^2 + 3*x^3 + 2*x^4)/(1-x^3)^2.
- A117900 (program): Expansion of (1 + 2*x + 4*x^2 + 4*x^3 + 2*x^4)/((1+x)*(1-x^3)^2).
- A117902 (program): Expansion of (1-x^2-2x^3)/(1-4x^3).
- A117904 (program): Number triangle [k<=n]*0^abs(L(C(n,2)/3) - L(C(k,2)/3)) where L(j/p) is the Legendre symbol of j and p.
- A117905 (program): Expansion of (1+2*x+2*x^2)/((1+x)*(1-x^3)^2).
- A117906 (program): Inverse of number triangle A117904.
- A117907 (program): Expansion of x + (1-x)^2/(1-x^6).
- A117908 (program): Chequered (or checkered) triangle for odd prime p=3.
- A117909 (program): Count, inserting 0 after every even number.
- A117910 (program): Expansion of (1 + x + x^2 + x^4)/((1-x^3)*(1-x^6)).
- A117916 (program): a(1) = 4; a(n) is smallest semiprime > 3*a(n-1).
- A117917 (program): a(n) = 3*a(n-1) + a(n-2) + n.
- A117918 (program): Difference row triangle of the Pell sequence.
- A117919 (program): Triangle read by rows: T(n, k) = 2^floor((k-1)/2)*binomial(n-1, k-1).
- A117926 (program): a(n) = n^floor(sqrt(n)).
- A117927 (program): a(n) = binomial(s(n), n) where s(n) = n-th semiprime.
- A117929 (program): Number of partitions of n into 2 distinct primes.
- A117935 (program): Triangle, row terms converge to the Pell sequence.
- A117938 (program): Triangle, columns generated from Lucas Polynomials.
- A117940 (program): a(0)=1, thereafter a(3n) = a(3n+1)/3 = a(n), a(3n+2)=0.
- A117942 (program): a(n) = a(3n) = -a(3n+1) = -a(3n+2)/2.
- A117943 (program): a(1) = 0, a(2) = 1; a(3n) = a(n); if every third term (a(3), a(6), a(9), …) is deleted, this gives back the original sequence.
- A117944 (program): Triangle related to powers of 3 partitions of n.
- A117945 (program): Triangle related to powers of 3 partitions of n.
- A117946 (program): a(3n)=0, a(3n+1)/a(1)=a(3n+2)/a(2)=A059151(n).
- A117947 (program): T(n,k)=L(C(n,k)/3) where L(j/p) is the Legendre symbol of j and p.
- A117948 (program): Sum of the divisors of pentagonal numbers.
- A117950 (program): a(n) = n^2 + 3.
- A117951 (program): a(n) = n^2 + 5.
- A117956 (program): Number of partitions of n into exactly 2 types of parts: one odd and one even.
- A117957 (program): Number of partitions of n into parts larger than 1 and congruent to 1 mod 4.
- A117960 (program): Triangular numbers with only odd digits.
- A117961 (program): Hexagonal numbers with prime indices.
- A117962 (program): Partial sums of hexagonal numbers with prime indices.
- A117963 (program): Antidiagonal sums of a Legendre-binomial triangle for p = 3.
- A117964 (program): A117963 mod 2.
- A117966 (program): Balanced ternary enumeration (based on balanced ternary representation) of integers; write n in ternary and then replace 2’s with (-1)’s.
- A117967 (program): Positive part of inverse of A117966; write n in balanced ternary and then replace (-1)’s with 2’s.
- A117968 (program): Negative part of inverse of A117966; write -n in balanced ternary and then replace (-1)’s with 2’s.
- A117972 (program): Numerator of zeta’(-2n), n >= 0.
- A117973 (program): a(n) = 2^(wt(n)+1), where wt() = A000120().
- A117976 (program): Legendre-binomial transform of 2^n for p=3.
- A117977 (program): Legendre-binomial transform of 3^n for p=3.
- A117980 (program): Legendre-binomial transform of (-1)^n for p=3.
- A117981 (program): A modified Legendre-binomial transform of 2^n for p=3.
- A117982 (program): Trisection of A117981.
- A117983 (program): A modified Legendre-binomial transform of 2^n for p=3.
- A117984 (program): A modified Legendre-binomial transform of 4^n for p=3.
- A117985 (program): Pentagonal numbers with only odd digits.
- A117989 (program): Number of partitions of n such that the least part occurs at least twice.
- A117993 (program): Heptagonal numbers with only odd digits.
- A117995 (program): Number of partitions of n in which both smallest and largest part occur only once.
- A117997 (program): Sum_{d|n} a(d) = n for n = 3^m (m >= 0) and for other n the sum is zero; i.e., the Möbius transform of [1, 0, 3, 0, 0, 0, 0, 0, 9, 0,…].
- A117998 (program): Decimal number generated by the binary bits of the n-th generation of the Rule 102 elementary cellular automaton.
- A117999 (program): Decimal number generated by the binary bits of the n-th generation of the Rule 110 elementary cellular automaton.
- A118000 (program): a(0) = 0. a(n) = a(n-1) + (smallest integer >= n which is coprime to a(n-1)).
- A118001 (program): a(n) = smallest integer >= n which is coprime to A118000(n-1). a(n) = A118000(n) - A118000(n-1).
- A118002 (program): a(0) = 0. a(n) = a(n-1) + (largest integer <= n which is coprime to a(n-1)).
- A118003 (program): a(n) = largest integer <= n which is coprime to A118002(n-1). a(n) = A118002(n) - A118002(n-1).
- A118004 (program): a(n) = 9^n - 4^n.
- A118005 (program): a(n) = ((-1)^n*5^(n+1) + 9^(n+1))/14.
- A118006 (program): Define a sequence of binary words by w(1) = 01 and w(n+1) = w(n)w(n)Reverse[w(n)]. Sequence gives the limiting word w(infinity).
- A118007 (program): Triangle, diagonals generated from Lucas polynomials.
- A118010 (program): Difference row triangle of A118009.
- A118011 (program): Complement of the Connell sequence (A001614); a(n) = 4*n - A001614(n).
- A118012 (program): a(n) = 4*A117384(n) - n; a self-inverse permutation of the natural numbers.
- A118013 (program): Triangle read by rows: T(n,k) = floor(n^2/k), 1<=k<=n.
- A118014 (program): Sum_{k=1..n} floor(n^2/k).
- A118015 (program): a(n) = floor(n^2/5).
- A118057 (program): a(n) = 8*n^2 - 4*n - 3.
- A118058 (program): a(n) = 49n^2 - 28n - 20.
- A118059 (program): 288*n^2 - 168*n - 119.
- A118060 (program): a(n) = 1681*n^2 - 984*n - 696.
- A118061 (program): 9800*n^2-5740*n-4059
- A118070 (program): Numbers with exactly one even decimal digit.
- A118071 (program): Primes which are the sum of a twin prime pair + 1.
- A118074 (program): Start with 1 and repeatedly reverse the digits and add 41 to get the next term.
- A118075 (program): Start with 1 and repeatedly reverse the digits and add 42 to get the next term.
- A118081 (program): Even numbers that can’t be represented as the sum of two odd composite numbers.
- A118083 (program): Number of partitions of n such that largest part k occurs at least floor(k/2) times.
- A118087 (program): Start with 1 and repeatedly reverse the digits and add 43 to get the next term.
- A118088 (program): a(0) = 0, a(n) = (1+2*0^(n mod 3))*a(floor(n/3)) + n mod 3.
- A118090 (program): Start with 1 and repeatedly reverse the digits and add 44 to get the next term.
- A118091 (program): Start with 1 and repeatedly reverse the digits and add 46 to get the next term.
- A118093 (program): Numbers of rooted hypermaps on the torus with n darts (darts are semi-edges in the particular case of ordinary maps).
- A118101 (program): Decimal representation of n-th iteration of the Rule 94 elementary cellular automaton starting with a single ON cell.
- A118102 (program): Triangle read by rows giving successive states of cellular automaton generated by “Rule 94” initiated with a single ON (black) cell.
- A118108 (program): Decimal representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single black cell.
- A118109 (program): Binary representation of n-th iteration of the Rule 54 elementary cellular automaton starting with a single black cell.
- A118110 (program): State of one-dimensional cellular automaton ‘sigma’ (Rule 150): 000,001,010,011,100,101,110,111 -> 0,1,1,0,1,0,0,1 at generation n, when started with a single ON cell, regarded as a binary number.
- A118111 (program): Binary representation of n-th iteration of the Rule 190 elementary cellular automaton starting with a single black cell.
- A118112 (program): a(n) = binomial(3n,n) mod (n+1).
- A118113 (program): Even Fibbinary numbers + 1; also 2*Fibbinary(n) + 1.
- A118114 (program): a(n) = binomial(3n,n) mod((n+1)(n+2)).
- A118115 (program): Partial sums of n concatenated n times.
- A118117 (program): Concatenate n F(n) times.
- A118124 (program): Triangle T(n,m) = (n+m)^2+n+m+41, read by rows.
- A118136 (program): 2 + (2*n! mod n+1).
- A118137 (program): Sum of decimal digits of two successive natural numbers.
- A118138 (program): Sum of factorials of prime factors, with multiplicity.
- A118139 (program): Numbers expressible as the sum of two triangular numbers in at least two different ways.
- A118140 (program): Index of A005846(n) in the primes.
- A118144 (program): Numbers of prime factors of l, where l is defined in A118534.
- A118145 (program): Start with 1 and repeatedly reverse the digits and add 47 to get the next term.
- A118146 (program): Start with 1 and repeatedly reverse the digits and add 49 to get the next term.
- A118147 (program): Start with 1 and repeatedly reverse the digits and add 50 to get the next term.
- A118148 (program): Start with 1 and repeatedly reverse the digits and add 51 to get the next term.
- A118149 (program): Start with 1 and repeatedly reverse the digits and add 52 to get the next term.
- A118150 (program): Start with 1 and repeatedly reverse the digits and add 53 to get the next term.
- A118151 (program): Start with 1 and repeatedly reverse the digits and add 54 to get the next term.
- A118152 (program): Start with 1 and repeatedly reverse the digits and add 56 to get the next term.
- A118153 (program): Start with 1 and repeatedly reverse the digits and add 57 to get the next term.
- A118154 (program): Start with 1 and repeatedly reverse the digits and add 58 to get the next term.
- A118155 (program): Start with 1 and repeatedly reverse the digits and add 59 to get the next term.
- A118156 (program): Start with 1 and repeatedly reverse the digits and add 61 to get the next term.
- A118157 (program): Start with 1 and repeatedly reverse the digits and add 62 to get the next term.
- A118158 (program): Start with 1 and repeatedly reverse the digits and add 63 to get the next term.
- A118159 (program): Start with 1 and repeatedly reverse the digits and add 64 to get the next term.
- A118160 (program): Start with 1 and repeatedly reverse the digits and add 48 to get the next term.
- A118161 (program): Start with 1 and repeatedly reverse the digits and add 55 to get the next term.
- A118162 (program): Start with 1 and repeatedly reverse the digits and add 60 to get the next term.
- A118163 (program): Start with 1 and repeatedly reverse the digits and add 65 to get the next term.
- A118170 (program): x for which abs(n^n-x!) is minimal for given n.
- A118171 (program): Decimal representation of n-th iteration of the Rule 158 elementary cellular automaton starting with a single black cell.
- A118173 (program): Decimal representation of n-th iteration of the Rule 188 elementary cellular automaton starting with a single black cell.
- A118175 (program): Binary representation of n-th iteration of the Rule 220 elementary cellular automaton starting with a single black cell.
- A118180 (program): Triangle T(n, k) = 3^(k*(n-k)), read by rows.
- A118181 (program): Row sums of triangle A118180: a(n) = Sum_{k=0..n} (3^k)^(n-k) for n>=0.
- A118182 (program): Antidiagonal sums of triangle A118180: a(n) = Sum_{k=0..[n/2]} (3^k)^(n-2*k) for n>=0.
- A118185 (program): Triangle T(n,k) = 4^(k*(n-k)) for n>=k>=0, read by rows.
- A118186 (program): Row sums of triangle A118185: a(n) = Sum_{k=0..n} 4^(k*(n-k)) for n>=0.
- A118187 (program): Antidiagonal sums of triangle A118185: a(n) = Sum_{k=0..[n/2]} 4^(k*(n-2*k)) for n>=0.
- A118190 (program): Triangle T(n,k) = 5^(k*(n-k)) for n >= k >= 0, read by rows.
- A118191 (program): Row sums of triangle A118190: a(n) = Sum_{k=0..n} 5^(k*(n-k)) for n>=0.
- A118192 (program): Antidiagonal sums of triangle A118190: a(n) = Sum_{k=0..floor(n/2)} 5^(k*(n-2*k)) for n>=0.
- A118200 (program): Start with 1 and repeatedly reverse the digits and add 66 to get the next term.
- A118205 (program): Euler transform of the negative of the Liouville function.
- A118206 (program): Euler transform of the Liouville function.
- A118207 (program): Expansion of Product_{k>=1}(1 + x^k)^lambda(k) where lambda(k) is the Liouville function, A008836.
- A118208 (program): G.f.: A(x) = product_{k>=1}(1 + x^k)^(-lambda(k)) where lambda(k) is the Liouville function, A008836.
- A118209 (program): Expansion of Sum_{k>=1} lambda(k) * k * x^k/(1 + x^k) where lambda(k) is the Liouville function, A008836.
- A118210 (program): Numerators of the coefficients of (x-1)(x-2)… in the interpolating polynomial through the first n primes.
- A118211 (program): Denominator of the coefficients of (x-1)(x-2)… in the interpolating polynomial through the first n primes.
- A118214 (program): Start with 1 and repeatedly reverse the digits and add 67 to get the next term.
- A118215 (program): Start with 1 and repeatedly reverse the digits and add 68 to get the next term.
- A118216 (program): Start with 1 and repeatedly reverse the digits and add 69 to get the next term.
- A118217 (program): Start with 1 and repeatedly reverse the digits and add 70 to get the next term.
- A118218 (program): Start with 1 and repeatedly reverse the digits and add 71 to get the next term.
- A118220 (program): Start with 1 and repeatedly reverse the digits and add 72 to get the next term.
- A118221 (program): Start with 1 and repeatedly reverse the digits and add 73 to get the next term.
- A118225 (program): Start with 1 and repeatedly reverse the digits and add 74 to get the next term.
- A118226 (program): Start with 1 and repeatedly reverse the digits and add 76 to get the next term.
- A118235 (