Sequences

From the game of Mousetrap.

Number of unreformed permutations of {1,...,n}.

Number of once reformable permutations of {1,2,...,n}.

Number of 4-level rooted trees with n leaves.

Number of 5-level rooted trees with n leaves.

Number of 5-leaf rooted trees with n levels.

Number of polynomial symmetric functions of matrix of order n under separate row and column permutations.

Number of symmetric polynomial functions of degree n of a symmetric matrix (of indefinitely large size) under joint row and column permutations. Also number of multigraphs with n edges (allowing loops) on an infinite set of nodes.

Number of independent polynomial invariants of matrix of order n.

Number of independent polynomial invariants of symmetric matrix of order n.

Arises in attempts to apply Engel's `probabilistic abacus' to random walks on infinite square grid.

Number of distinct degree sequences among all connected graphs with n nodes.

Number of graphical partitions of biconnected graphs with n nodes.

Triangle a(n,k) of number of M-sequences read by antidiagonals.

Even minus odd extensions of truncated 3 X 2n grid diagram.

Number of spanning trees of Aztec diamonds of order n.

Number of spanning trees of quarter Aztec diamonds of order n.

Number of 2n-bead black-white strings with n black beads and fundamental period 2n.

5th binary partition function.

6th binary partition function.

7th binary partition function.

a(n) = a(floor(n/2)) + a(floor(n/3)) + a(floor(n/6)), with a(0) = 1.

Period of decimal representation of 1/n.

Period of binary representation of 1/n. Also, multiplicative order of 2 modulo the odd part of n (= A000265(n)).

Period of repeating digits of 1/n in base 3.

Period of base 4 representation of 1/n.

Period of repeating digits of 1/n in base 5.

Period of repeating digits of 1/n in base 6.

Period of repeating digits of 1/n in base 7.

Period of repeating digits of 1/n in base 8.

Period of repeating digits of 1/n in base 9.

a(n) = prime(n)*...*prime(m), the least product of consecutive primes which is abundant.

a(n) = n*(4*n+1).

Number of achiral polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4} (or polycubes).

Expansion of (1+6*x)/(1-4*x)^(7/2).

a(n) = n OR n^2 (applied to binary expansions).

Number of ways for n-3 nonintersecting loops to cross a line 2n times.

Number of nonnegative integer points (p_1,p_2,...,p_n) in polytope defined by p_0 = p_{n+1} = 0, 2p_i - (p_{i+1} + p_{i-1}) <= 2, p_i >= 0, i=1,...,n. Number of score sequences in a chess tournament with n+1 players (with 3 outcomes for each game).

Number of self-converse oriented trees with n nodes.

Numbers k such that k!! - 1 is prime.

Nonnegative integers n such that n^2*(n+1)*(2*n+1)^2*(7*n+1)/36 is a square.

Even bisection of A007750.

Odd bisection of A007750.

a(n) = Sum_{k=0..n-1} binomial(a(k)^2, k).

Array (a frieze pattern) defined by a(n,k) = (a(n-1,k)*a(n-1,k+1) - 1) / a(n-2,k+1), read by antidiagonals.

Smallest number m such that the trajectory of m under iteration of Euler's totient function phi(n) [A000010] contains exactly n distinct numbers, including m and the fixed point.

Number of irreducible indecomposable spherical curves with n crossings (only ordinary double points), the circle is oriented, the sphere is not oriented (OU case).

Dwork-Kontsevich sequence evaluated at 2*n.

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

Knopfmacher expansion of sqrt(2): a(2n) = 2*(a(2n-1) + 1)^2 - 1, a(2n+1) = 2*(a(2n)^2 - 1).