Sequences

Glaisher's function H'(4n+1) (18 squares version).

Glaisher's function V(n).

Glaisher's function U(n).

Glaisher's function J(n) (18 squares version).

Glaisher's function theta(n) (18 squares version).

Glaisher's function T_1(n).

Reduced totient function (divided by 2).

Weight distribution of Cheng-Sloane [ 32,17,8 ] code.

a(n) = n*phi(n).

Number of 2-colored patterns on an n X n board.

Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4).

Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*(1-x^4)).

Number of partitions of at most n into at most 5 parts.

Expansion of 1/((1-x)^4*(1+x)).

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

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

Expansion of 1/((1-x)^3 (1-x^2)^2 (1-x^3) (1-x^4)).

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

Number of permutations of length n without 3-sequences.

Number of permutations of length n with one 3-sequence.

Number of permutations of length n with two 3-sequences.

Number of circuits of nullity n.

Erroneous version of A377569.

Related to discordant permutations.

From discordant permutations.

Number of partitions of n into 4 squares.

Number of ways of writing n as an unordered sum of at most 3 nonzero triangular numbers.

Number of partitions of n into not more than 5 pentagonal numbers.

a(n) = (number of nonisomorphic nontransitive prime tournaments on n nodes) - Moebius(n).

Numerators of expansion of Jacobi nome q in parameter m.

Numbers k such that (k^2 + k + 1)/3 is prime.

Numbers k such that (k^2 + k + 1)/7 is prime.

Numbers k such that (k^2 + k + 1)/13 is prime.

Numbers k such that (k^2 + k + 1)/19 is prime.

Numbers k such that (k^2 + k + 1)/21 is prime.

Quartan primes: primes of the form x^4 + y^4, x > 0, y > 0.

Half-quartan primes: primes of the form p = (x^4 + y^4)/2.

Sextan primes: p = (x^6 + y^6)/(x^2 + y^2).

A variant of the cuban primes: primes p = (x^3 - y^3)/(x - y) where x = y + 2.

Quintan primes: p = (x^5 - y^5)/(x - y).

Quintan primes: p = (x^5 + y^5)/(x + y).

Dates at 16-day intervals starting at the beginning of a leap year.

Theta series of Kleinian lattice Z[(1 + sqrt(-7))/ 2] in 1 complex (or 2 real) dimensions.

Expansion of (theta_3(z)*theta_3(7z)+theta_2(z)*theta_2(7z))^3.

Number of ways of writing n as a sum of at most two nonzero squares, where order matters; also (number of divisors of n of form 4m+1) - (number of divisors of form 4m+3).

Expansion of Product_{i >= 1} (1 - q^i)*(1 - q^(7*i)).

Expansion of (eta(q) * eta(q^7))^3 in powers of q.

Numerators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).

a(0) = a(1) = 1; for n > 0, a(n+1) = a(n)*(a(0) + ... + a(n-1)) + a(n)*(a(n) + 1)/2.

a(n) = 2*sigma(n) - 1.