Sequences

A Lucas-Lehmer sequence: a(0) = 4; for n>0, a(n) = a(n-1)^2 - 2.

Number of permutations of up to n kinds of objects, where each kind of object can occur at most two times.

Maximum number of cubes of side length 2 that can be packed into a 3-dimensional torus of side length 2*n+1.

E.g.f. 1 + x*exp(x) + x^2*exp(2*x).

Expansion of e.g.f.: 1 + x*exp(x) + x^2*exp(2*x) + x^3*exp(3*x).

Numbers that occur 5 or more times in Pascal's triangle.

Number of occurrences of n as an entry in rows <= n of Pascal's triangle (A007318).

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

Number of distinct values taken by 3^3^...^3 (with n 3's and parentheses inserted in all possible ways).

Number of distinct values taken by 4^4^...^4 (with n 4's and parentheses inserted in all possible ways).

Largest prime factor of the "repunit" number 11...1 (cf. A002275).

Largest prime factor of 10^n + 1.

Length of shortest (or optimal) Golomb ruler with n marks.

"Length" of aliquot sequence for n.

Number of acyclic digraphs (or DAGs) with n labeled nodes.

Number of n-node labeled acyclic digraphs with 1 out-point.

Number of n-node labeled acyclic digraphs with 2 out-points.

Number of weakly connected digraphs with n labeled nodes.

Number of digraphs on n labeled nodes with a source.

Number of unilaterally connected digraphs with n labeled nodes.

Number of strongly connected digraphs with n labeled nodes.

Denominators of expansion of Fresnel integral S(z).

Smallest integer m such that the product of every 3 consecutive integers > m has a prime factor > prime(n).

Smallest integer m such that the product of every 4 consecutive integers > m has a prime factor > prime(n).

Sylvester's problem: minimal number of ordinary lines through n points in the plane.

Maximal number of 3-tree rows in n-tree orchard problem.

Number of simplicial arrangements of n lines in the plane (the lines do not pass through a common point, all cells are triangles).

Smallest number of complexity n: smallest number requiring n 1's to build using +, * and ^.

Dimensions of split simple Lie algebras over any field of characteristic zero.

Maximal number of prime implicants of a Boolean function of n variables.

Highest degree of an irreducible representation of symmetric group S_n of degree n.

Number of vacuously transitive relations on n nodes up to isomorphism.

Number of directed Hamiltonian cycles (or Gray codes) on n-cube.

Number of Hamiltonian paths (or Gray codes) on n-cube with a marked starting node.

For n > 4, a(n) is the least integer > a(n-1) with precisely two representations a(n) = a(i) + a(j), 1 <= i < j < n; and a(n) = n for n=1..4.

a(n) (n>6) is least integer > a(n-1) with precisely three representations a(n) = a(i) + a(j), 1 <= i < j < n, a(n) = n for n=1..6.

Product of first n Catalan numbers.

a(n) = Catalan(n) * Product_{k = 0..n-1} a(k).

a(n+1) = n*a(n) - (-1)^n.

Number of connected Eulerian graphs with n unlabeled nodes.

Number of primitive sublattices of index n in hexagonal lattice: triples x,y,z from Z/nZ with x+y+z = 0, discarding any triple that can be obtained from another by multiplying by a unit and permuting.

Number of inequivalent sublattices of index n in hexagonal lattice, where two sublattices are equivalent if they are related by a rotation or reflection preserving the hexagonal lattice.

Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m).

Order of orthogonal group O(n, GF(2)).

Erroneous version of A001011 ("folding a strip of stamps").

Number of connected graphs, up to homeomorphism, that can be drawn in the plane using unit-length edges.

n appears n+1 times. Also the array A(n,k) = n+k (n >= 0, k >= 0) read by antidiagonals. Also inverse of triangular numbers.

n appears n - 1 times.

Duplicate of A000194.

k appears 2k-1 times. Also, square root of n, rounded up.