A320150 Number of integer solutions to a^2 + 2*b^2 + 2*c^2 + 3*d^2 = n.

1, 2, 4, 10, 10, 16, 24, 12, 28, 26, 8, 48, 30, 28, 72, 32, 34, 64, 28, 36, 80, 60, 72, 96, 72, 42, 56, 82, 36, 112, 120, 60, 124, 96, 32, 96, 130, 76, 216, 140, 56, 160, 48, 84, 144, 112, 144, 192, 150, 86, 84, 128, 140, 208, 240, 96, 216, 180, 56, 240, 96, 124, 360

Offset

0,2

Comments

a(n) > 0 for n >= 0.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Simon Plouffe, Numbers in the base e^Pi, arXiv:2509.15609 [math.NT], 2025. See p. 21/24, marked 214.

Formula

G.f.: theta_3(q) * theta_3(q^2)^2 * theta_3(q^3).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/48) * 3^(3/4) * Gamma(2/3) * 2^(1/4) * Gamma(5/8)^3 * (2+sqrt(2))^(3/2) / (3^(1/2)-1) / (sqrt(2)-1) / Pi / Gamma(7/8)^3 / Gamma(11/12) = A389053. - Simon Plouffe, Sep 22 2025

Crossrefs

Cf. A320149, A236924, A320151, A320152, A320153.

Sequence in context: A356603 A097211 A092945 * A056376 A185372 A186757

Adjacent sequences: A320147 A320148 A320149 * A320151 A320152 A320153

Keyword

nonn

Author

Seiichi Manyama, Oct 06 2018

Status

approved