Sequences

a(0) = 1, a(n) = 21*n^2 + 2 for n>0.

a(0) = 1, a(n) = 22*n^2 + 2 for n>0.

a(0) = 1, a(n) = 23*n^2 + 2 for n>0.

a(0) = 1, a(n) = 24*n^2 + 2 for n>0.

a(0) = 1, a(n) = 25*n^2 + 2 for n > 0.

a(0) = 1, a(n) = 26*n^2 + 2 for n>0.

a(0) = 1, a(n) = 27*n^2 + 2 for n>0.

a(0) = 1, a(n) = 28*n^2 + 2 for n>0.

a(0) = 1, a(n) = 29*n^2 + 2 for n>0.

a(0) = 1, a(n) = 31*n^2 + 2 for n>0.

a(0) = 1, a(n) = 32*n^2 + 2 for n > 0.

a(0) = 1, a(n) = 40*n^2 + 2 for n>0.

a(0) = 1, a(n) = 42*n^2 + 2 for n>0.

Coordination sequence for squashed {D_5}* lattice, perhaps the smallest example of a "non-superficial" lattice.

Crystal ball sequence for squashed {D_5}^* lattice, perhaps the smallest example of a "non-superficial" lattice.

Triangle read by rows: number of permutations of 1..n by length of longest run.

Triangle read by rows: T(n,k) is the number of permutations of [n] having k consecutive ascending pairs (0 <= k <= n-1).

Triangle read by rows: T(n,k) is one-half the number of permutations of length n with exactly n-k rising or falling successions, for n >= 1, 1 <= k <= n. T(1,1) = 1 by convention.

Irregular triangle read by rows: T(n,k) (n>=1, 0 <= k <= floor(n/2)) = number of permutations of 1..n with exactly floor(n/2) - k runs of consecutive pairs up.

Irregular triangle read by rows: T(n,k) (n >= 1, 0 <= k <= [n/2]) = number of permutations of 1..n with [n/2]-k runs of consecutive pairs up and down (divided by 2).

Weight distribution of any one of the five doubly-even binary [32,16,8] codes (quadratic residue, Reed-Muller, etc.).

Weight distribution of binary (16,256,6) nonlinear Nordstrom-Robinson code.

A thinks of x in set M; B asks questions: is x in T?; A may lie once but only when true answer is Yes; a(n) is maximal size of M such that B can determine x with <= n questions.

Numbers k such that gcd(k^17 + 9, (k+1)^17 + 9) > 1.

a(n) = 2*3^(2*n)-3^n.

Sum of 2^n, ..., 2^(n+1) - 1.

Numbers n such that gcd(n^5 + 5, (n+1)^5 + 5) > 1.

Number of letters in n (in Czech).

High-temperature expansion of Ising model susceptibility chi_2 for square lattice.

High-temperature expansion of Ising model susceptibility chi_2 for cubic lattice.

High-temperature expansion of Ising model susceptibility chi_2 for 4-d cubic lattice.

High-temperature expansion of susceptibility mu_2 for square lattice.

High-temperature expansion of susceptibility mu_2 for cubic lattice.

High-temperature expansion of susceptibility mu_2 for 4-d cubic lattice.

High-temperature expansion of Ising model susceptibility chi_4 for square lattice.

High-temperature expansion of Ising model susceptibility chi_4 for cubic lattice.

High-temperature expansion of Ising model susceptibility chi_4 for 4-d cubic lattice.

Triangle of Fibonomial coefficients, read by rows.

Second-order Fibonacci numbers.

a(n) = (2n)!.

Characteristic function of primes: 1 if n is prime, else 0.

Characteristic function of squares: a(n) = 1 if n is a square, otherwise 0.

a(n) = 4^n*(2*n+1)!*(n!)^2/(n+1).

a(n) = 1 if n is a triangular number, otherwise 0.

1 if n is a prime power p^k (k >= 0), otherwise 0.

Characteristic function of Fibonacci numbers: a(n) = 1 if n is a Fibonacci number, otherwise 0.

a(n) = 1 if n is a cube, else 0.

1 if n is a Catalan number else 0.

Another version of the 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 0's and 1's.

Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 0 and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 0's and 1's.