Sequences

Maximal number of points in PG(2,q) with at most 3 on a line (next term is 21 or 22).

Number of n-step spiral self-avoiding walks on hexagonal lattice, where at each step one may continue in same direction or make turn of 2*Pi/3 counterclockwise.

Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 3, where equivalence is defined by row and column permutations.

Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 4, where equivalence is defined by row and column permutations. Also number of isomorphism classes of bicolored quartic bipartite graphs, where isomorphism cannot exchange the colors.

Eulerian numbers (Euler's triangle: column k=6 of A008292, column k=5 of A173018).

a(n) = (2n)!(2n+1)!/n!^4, or equally (2n+1)*binomial(2n,n)^2.

Number of equivalence classes of n X n matrices over {0,1} with rows and columns summing to 5, where equivalence is defined by row and column permutations. Isomorphism classes of bicolored 5-regular bipartite graphs, where isomorphism cannot exchange the colors.

Number of permutations of length n with exactly three valleys.

Generalized tangent numbers d_(n,4).

Number of equivalence classes of nonzero regular 0-1 matrices of order n.

Nearest integer to log_10(n).

Coefficients of modular function j as power series in q = e^(2 Pi i t). Another name is the elliptic modular invariant J(tau).

Total number of ordered k-tuples (k=0..n) of distinct elements from an n-element set: a(n) = Sum_{k=0..n} n!/k!.

a(n) = floor(log_2(n)).

Number of rooted trees with n nodes, 2 of which are labeled.

Number of partially labeled rooted trees with n nodes (4 of which are labeled).

Number of partially labeled trees with n nodes (5 of which are labeled).

Series-parallel numbers.

Number of types of Latin squares of order n. Equivalently, number of nonisomorphic 1-factorizations of K_{n,n}.

Powers of rooted tree enumerator.

Let p(n, s, x) be predicate that number of occurrences of s's in x >= 2*n - the length of the longest sequence of s's in x. Then a(n)=#{x in {0,1}* | x ends in 0 and p(n,0,x) and (there is no prefix y of x such that p(n,0,y) or p(n,1,y))}.

From area of cyclic polygon of 2n + 1 sides.

Number of Hamiltonian paths from NW to SW corners in an n X n grid.

a(0)=1; a(n) = 10^n + 1, n >= 1.

Numbers that are not the sum of 4 nonzero squares.

Card matching: coefficients B[n,2] of t^2 in the reduced hit polynomial A[n,n,n](t).

Number of 3-line Latin rectangles.

Sum of first n cubes; or n-th triangular number squared.

Sum of fourth powers: 0^4 + 1^4 + ... + n^4.

Sum of 5th powers: 0^5 + 1^5 + 2^5 + ... + n^5.

Sum of 6th powers: 0^6 + 1^6 + 2^6 + ... + n^6.

Sum of 7th powers: 1^7 + 2^7 + ... + n^7.

Sum of 8th powers: 1^8 + 2^8 + ... + n^8.

Number of inequivalent ways to color vertices of a cube using at most n colors.

Number of permutations of length n by rises.

Number of ways of n-coloring a dodecahedron.

First occurrence of n consecutive numbers that take same number of steps to reach 1 in 3x+1 problem.

Number of steps to reach 1 in sequence A000546.

Squares that are not the sum of 2 nonzero squares.

Numbers that are the sum of 2 squares but not sum of 3 nonzero squares.

Number of trees of diameter 7.

Number of labeled rooted trees of height 2 with n nodes.

Number of labeled rooted trees of height 3 with n nodes.

Number of labeled rooted trees of height 4 with n nodes.

Number of labeled trees of diameter 3 with n nodes.

Number of labeled trees of diameter 4 with n nodes.

Expansion of exp(-x) / (1 - exp(x) + exp(-x)).

Expansion of e.g.f. 1/(1 - 2*sinh(x)).

Generalized Stirling numbers of second kind.

Generalized Stirling numbers of second kind.