Sequences

Image of n after 3k iterates of '3x+1' map (k large).

Percolation series for directed square lattice.

Percolation series for directed square lattice.

Convolve natural numbers with characteristic function of triangular numbers.

Continued fraction for Sum_{n>=0} 1/4^(2^n).

Continued fraction for 4^5*Sum_{n>=0} 1/4^(2^n).

Continued fraction expansion of C = 2*Sum_{n>=0} 1/2^(2^n).

Continued fraction for Sum_{n>=0} (-1)^n/3^(2^n).

Number of rooted planar maps with 4 faces and n vertices and no isthmuses.

Number of rooted toroidal maps with 2 faces, n vertices and no isthmuses.

Number of tree-rooted planar maps with 3 faces and n vertices and no isthmuses.

Number of tree-rooted planar maps with 4 faces and n vertices and no isthmuses.

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

a(n) = binomial(n,2)!/n!.

Related to Ramsey numbers.

Number of 2-tournaments on n nodes.

From relations between Siegel theta series.

Number of partitions of n with at least 1 odd and 1 even part.

a(n) = a(n-1) + a(n-2) + F(n) - 1, a(0) = a(1) = 0, where F() = Fibonacci numbers A000045.

From variance of Fibonacci search.

De Bruijn's S(3,n): (3n)!/(n!)^3.

Euler characteristics of polytopes.

Euler characteristics of polytopes.

a(n) = Fibonacci(n)*2^n + 1.

a(n) = n*(n + 1)*(n^2 - 3*n + 5)/6.

a(n) = (2^(2^n + 1) + 1)/3.

a(n) = largest prime factor of n^n - 1.

Denominators of greedy Egyptian fraction for square root of 2.

Numbers n such that n! has a square number of digits.

Numbers k such that k-6, k, and k+6 are primes.

a(1) = 1, a(2) = 0; for n > 2, a(n) = n*Fibonacci(n-2) (with the convention Fibonacci(0)=0, Fibonacci(1)=1).

Generalized Lucas numbers.

Generalized Lucas numbers.

Generalized Lucas numbers.

Number of possible chess games at the end of the n-th ply plus number of games that terminate (i.e., mate) in fewer than n plies.

Real part of (1 + 2*i)^n, where i is sqrt(-1).

Imaginary part of (1+2i)^n.

a(n) = 3*a(n-1) + a(n-2) with a(0) = 2, a(1) = 3.

a(n) = a(n-1) + a(n-3) + a(n-4), a(0) = a(1) = a(2) = 1, a(3) = 2.

Number of restricted circular combinations.

Restricted combinations.

Expansion of (1+x^2) / ( (1-x)^2 * (1-x^3)^2 ).

Related to Fibonacci numbers.

a(n) = n*(n+1)*(n+8)/6.

Coefficient of x^4 in (1-x-x^2)^(-n).

Number of partitions of an n-set into boxes of size >2.

Number of n X n binary matrices with no 2 adjacent 1's, or number of configurations of non-attacking princes on an n X n board, where a "prince" attacks the four adjacent (non-diagonal) squares. Also number of independent vertex sets in an n X n grid.

a(n+1) = a(n) + sum of digits of a(n), with a(1)=7.

a(n+1) = a(n)-th composite number, with a(0) = 1.

Cald's sequence: a(n+1) = a(n) - prime(n) if that value is positive and new, otherwise a(n) + prime(n) if new, otherwise 0; start with a(1)=1.