A320138 Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 3*d^2 = n.

1, 2, 2, 8, 10, 8, 24, 16, 10, 38, 8, 12, 48, 8, 32, 64, 26, 36, 70, 28, 24, 80, 28, 48, 96, 42, 40, 76, 48, 24, 112, 64, 58, 160, 68, 32, 126, 56, 44, 192, 56, 84, 176, 44, 60, 88, 80, 96, 208, 114, 74, 176, 72, 72, 172, 80, 112, 288, 88, 76, 224, 72, 112, 304, 90, 96

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. 19/24, marked 212.

Formula

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

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/128) * Gamma(2/3)^2 * Gamma(5/8)^6 * Gamma(7/12)^2 * (7 * sqrt(2)+10) * (2+3^(1/2)) * sqrt(2) * (2+sqrt(2))^(1/2) / Pi^3 / Gamma(7/8)^6 = A389049. - Simon Plouffe, Sep 22 2025

Crossrefs

Cf. A320139, A320140, A033712, A320188, A320189, A320190, A320191.

Sequence in context: A121094 A029595 A343447 * A325102 A389536 A275436

Adjacent sequences: A320135 A320136 A320137 * A320139 A320140 A320141

Keyword

nonn

Author

Seiichi Manyama, Oct 06 2018

Status

approved