A236926 Number of integer solutions to a^2 + b^2 + c^2 + 3*d^2 = n.

1, 6, 12, 10, 18, 48, 40, 12, 60, 78, 24, 48, 70, 84, 120, 32, 66, 192, 84, 36, 144, 180, 120, 96, 136, 126, 168, 82, 84, 336, 200, 60, 252, 288, 96, 96, 234, 228, 360, 140, 120, 480, 144, 84, 336, 336, 240, 192, 310, 258, 252, 128, 252, 624, 400, 96, 408, 540, 168

Offset

0,2

Links

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

J. Liouville, Sur La Forme x^2+y^2+z^2+3t^2, Journal de Mathématiques Pures et appliquées (2) 8, 193-204 (1863).

Formula

G.f.: theta_3(q)^3*theta_3(q^3), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/8) * sqrt(Pi) * 3^(1/2) * Gamma(2/3) * Gamma(11/12)^2 * Gamma(7/12)^3 * sqrt(2) * (1+3^(1/2)) / Gamma(3/4)^9 = A388822. - Simon Plouffe, Sep 18 2025

Maple

See A236924.

Crossrefs

Cf. A236924.

For number of solutions to a^2+b^2+c^2+k*d^2=n for k=1,2,3,4,5,6,7,8,12 see A000118, A236928, A236926, A236923, A236930, A236931, A236932, A236927, A236933.

Sequence in context: A360719 A365787 A315566 * A134383 A366144 A272305

Adjacent sequences: A236923 A236924 A236925 * A236927 A236928 A236929

Keyword

nonn

Author

N. J. A. Sloane, Feb 15 2014

Status

approved