A320147 - OEIS

A320147

Number of integer solutions to a^2 + b^2 + 3*c^2 + 5*d^2 = n.

1, 4, 4, 2, 12, 18, 8, 16, 24, 28, 40, 8, 26, 72, 16, 16, 44, 44, 68, 24, 34, 80, 72, 28, 40, 124, 40, 50, 112, 56, 80, 40, 76, 144, 120, 32, 84, 216, 24, 40, 136, 80, 160, 88, 56, 154, 88, 28, 158, 228, 100, 48, 216, 172, 80, 104, 80, 300, 280, 40, 112, 248, 120, 112

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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. 20/24, marked 213.

FORMULA

G.f.: theta_3(q)^2 * theta_3(q^3) * theta_3(q^5).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/320) * 2^(1/10) * 5^(1/4) * Gamma(2/3) * Gamma(7/12) * Gamma(3/5)^2 * Gamma(7/10) * (1+3^(1/2)) * (5+5^(1/2)) * (5^(1/2)+1)^2 / sqrt(Pi) / Gamma(3/4)^5 / Gamma(9/10) = A389051. - Simon Plouffe, Sep 22 2025

CROSSREFS

Cf. A034896, A272364, A320148.

Sequence in context: A224821 A034933 A320148 * A369410 A272364 A388418

Adjacent sequences: A320144 A320145 A320146 * A320148 A320149 A320150

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Oct 06 2018

STATUS

approved