Sequences

Smallest starting prime for n consecutive primes in arithmetic progression.

Number of intersections of diagonals in the interior of a regular n-gon.

Balanced primes (of order one): primes which are the average of the previous prime and the following prime.

(2*n)!-Sum ((-1)^(i+1)*binomial(n,i)*2^i*(2*i-1)!,i=1..n).

Icosahedral numbers: a(n) = n*(5*n^2 - 5*n + 2)/2.

Number of ways to color vertices of a hexagon using <= n colors, allowing only rotations.

Dodecahedral numbers: a(n) = n*(3*n - 1)*(3*n - 2)/2.

Emirps (primes whose reversal is a different prime).

Denominators of generalized Bernoulli numbers.

Numerators of generalized Bernoulli numbers.

From trees with valency <= 3.

Expansion of q*Product_{k>=1} (1-q^k)^2*(1-q^(11*k))^2.

Numerators of an asymptotic expansion for the number of forests on n nodes (A001858).

Denominators of an asymptotic expansion for the number of forests on n nodes (A001858).

Number of domino n-tuples.

Number of primitive (aperiodic, or Lyndon) asymmetric rhythm cycles: ones having no nontrivial shift automorphism.

Primitive repfigit numbers.

Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.

Triangular numbers plus quarter squares: n*(n+1)/2 + floor(n^2/4) (i.e., A000217(n) + A002620(n)).

a(n) = Sum_{k=1..n-1} gcd(n,k).

a(n) = Sum_{k=1..n-1} lcm(k,n-k).

a(n) = Sum_{k=1..n-1} (k AND n-k).

a(n) = Sum_{k=1..n-1} k XOR n-k.

a(n) = Sum_{k=1..n-1} (k OR n-k).

If n mod 2 = 0 then n*(n^2-4)/12 else n*(n^2-1)/12.

Egyptian fractions: number of solutions to 1 = 1/x_1 + ... + 1/x_n in positive integers x_1 < ... < x_n.

a(n) = Sum_{k=1..n} floor((2*n-1)/(2*k+1)).

a(n) = 3*2^(2*n)*(3*n)!/((2*n)!*n!).

a(n) = 4^n*(3*n)!/((2*n)!*n!).

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

a(n) = Sum_{k=1..n} ceiling(n/k).

a(n) = Sum_{k=1..n} nearest integer to n/k (if n/k is midway between two numbers take the smaller).

a(n) = 10*n^3 - 6*n^2.

Least number which is side of n Pythagorean triples.

A Beatty sequence: floor(n*(1 + 1/e)).

a(n) = (n+2)!/4 + n!/2.

Numbers k such that (2^(2k+1) - 2^(k+1) + 1)/5 is prime.

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

Numbers n such that 2^(2n+1) - 2^(n+1) + 1 is a prime.

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

Total number of triangles visible in regular n-gon with all diagonals drawn.

Numbers k such that k, k+1, k+2 and k+3 have the same number of divisors.

a(n) is the number of hierarchical models on n unlabeled factors or variables with linear terms forced.

Generalized Fibonacci numbers.

Generalized Fibonacci numbers.

Number of modes of connections of 2n points.

Nonperiodic autocorrelation functions of length n.

Number of labeled connected rooted trivalent graphs with 2n nodes.

Number of n-node graphs not determined by their spectrum.

Number of cyclic neofields of order n.