A320126 Number of integer solutions to a^2 + b^2 + 2*c^2 + 5*d^2 = n.

1, 4, 6, 8, 12, 10, 16, 28, 22, 36, 40, 24, 56, 36, 24, 64, 28, 64, 78, 24, 58, 48, 68, 92, 80, 92, 72, 112, 92, 48, 104, 96, 118, 176, 64, 48, 124, 84, 148, 160, 104, 120, 176, 120, 72, 146, 88, 204, 216, 124, 126, 96, 180, 148, 224, 188, 120, 304, 216, 120, 224, 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)^2 * theta_3(q^2) * theta_3(q^5).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/6400) * sqrt(2) * 5^(3/4) * Gamma(5/8)^4 * (3+2 * sqrt(2)) * (5-5^(1/2))^(3/2) * (2+sqrt(2))^(1/2) * (5^(1/2)+1)^3 / Pi / Gamma(7/8)^4 = A389048. - Simon Plouffe, Sep 22 2025

Crossrefs

Cf. A097057, A320124, A320125, A320127, A320128, A389048.

Sequence in context: A320127 A352374 A110646 * A234520 A275789 A031359

Adjacent sequences: A320123 A320124 A320125 * A320127 A320128 A320129

Keyword

nonn

Author

Seiichi Manyama, Oct 06 2018

Status

approved