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

Author

N. J. A. Sloane, Feb 15 2014

Keywords

• nonn

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).

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.

Programs

Maple

See A236924.

Formula

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