A320124 Number of integer solutions to a^2 + b^2 + 2*c^2 + 3*d^2 = n.

1, 4, 6, 10, 20, 20, 24, 40, 22, 28, 56, 20, 50, 80, 28, 80, 84, 32, 78, 80, 68, 100, 120, 80, 88, 124, 56, 82, 136, 100, 140, 200, 86, 80, 192, 72, 140, 240, 120, 200, 248, 80, 112, 176, 100, 260, 224, 160, 210, 172, 186, 128, 272, 180, 240, 400, 124, 200, 280, 116

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)^2 * theta_3(q^2) * theta_3(q^3).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/128) * 2^(3/4) * Gamma(2/3) * Gamma(5/8)^5 * Gamma(7/12) * (2+sqrt(2))^3 * (1+3^(1/2)) / Gamma(7/8)^5 / Pi^2 = A389047. - Simon Plouffe, Sep 22 2025

Crossrefs

Cf. A097057, A320125, A320126, A320127, A320128.

Sequence in context: A096817 A220033 A277455 * A185913 A243119 A277343

Adjacent sequences: A320121 A320122 A320123 * A320125 A320126 A320127

Keyword

nonn

Author

Seiichi Manyama, Oct 06 2018

Status

approved