Sequences

Local stops on New York City A line subway.

Number of trees with n unlabeled nodes.

Order of the group SL(2,Z_n).

Primes that divide at least one term in every Fibonacci sequence.

Sylvester's sequence: a(n+1) = a(n)^2 - a(n) + 1, with a(0) = 2.

Numbers k such that (2k)^4 + 1 is prime.

Number of signed trees with n nodes.

Generalized tangent numbers d(n,1).

A Beatty sequence: a(n) = floor(n/(e-2)).

Symmetrical dissections of an n-gon.

Partial sums of (unordered) ways of making change for n cents using coins of 1, 2, 5, 10 cents.

-1 + number of partitions of n.

Smallest number of vertices in trivalent graph with girth (shortest cycle) = n.

Number of positive integers <= 2^n of form x^2 + 2 y^2.

Numbers k such that k^4 + 1 is prime.

Odious numbers: numbers with an odd number of 1's in their binary expansion.

a(n) = Sum_{k=0..n} p(k) where p(k) = number of partitions of k (A000041).

a(n) = Fibonacci(n) - 1.

Number of positive integers <= 2^n of form x^2 + 4 y^2.

Tribonacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) for n >= 3 with a(0) = a(1) = 0 and a(2) = 1.

Number of odd integers <= 2^n of form x^2 + y^2.

Number of positive integers <= 2^n of form 2*x^2 + 3*y^2.

Number of integers <= 2^n of form 4 x^2 + 4 x y + 5 y^2.

Number of positive integers <= 2^n of form x^2 + 6 y^2.

Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) for n >= 4 with a(0) = a(1) = a(2) = 0 and a(3) = 1.

Number of nonisomorphic minimal triangle graphs.

Number of unlabeled rooted trees with n nodes (or connected functions with a fixed point).

a(n) = n^2*Product_{p|n} (1 + 1/p).

Number of mixed Husimi trees with n nodes; or polygonal cacti with bridges.

Number of series-parallel networks with n unlabeled edges. Also called yoke-chains by Cayley and MacMahon.

Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.

Number of solutions to x^2 - x + 1 == 0 (mod n).

Number of unrooted nonseparable planar maps with n edges and a distinguished face.

Number of simple graphs on n unlabeled nodes.

Number of solutions to x^2 + 1 == 0 (mod n).

Expansion of e.g.f. exp((-x^3)/3)/(1-x).

Multiplicative with a(2^e) = 2 for k >= 1; a(3) = 2, a(3^e) = 0 for k >= 2; a(p^e) = 0 if p > 3 and p == -1 (mod 3); a(p^e) = 2 if p > 3 and p == 1 (mod 3).

Let A(n) = #{(i,j,k): i^2 + j^2 + k^2 <= n}, V(n) = (4/3)Pi*n^(3/2), P(n) = A(n) - V(n); sequence gives values of n where |P(n)| sets a new record.

a(n) = floor(n^(3/2)).

Number of trees of diameter 4.

Number of fixed points of Gamma_0 (n) of type i.

a(n) = n*(n+3)/2.

Number of partitions of n if there are two kinds of 1's and two kinds of 2's.

Number of partitions of n if there are two kinds of 1, two kinds of 2 and two kinds of 3.

Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); sequence gives values of n where |P(n)| sets a new record.

a(n) is the number of compositions of n in which the maximal part is 3.

Record gaps between primes (upper end) (compare A002386, which gives lower ends of these gaps).

a(n) = number of compositions of n in which the maximum part size is 4.

Number of n-node triangulations of sphere in which every node has degree >= 4.

Number of n-celled free polyominoes without holes.