A236928 Number of integer solutions to a^2 + b^2 + c^2 + 2*d^2 = n.

1, 6, 14, 20, 30, 40, 36, 48, 62, 42, 72, 100, 68, 120, 112, 48, 126, 108, 98, 180, 136, 160, 180, 144, 132, 126, 216, 200, 240, 280, 112, 192, 254, 120, 252, 320, 210, 360, 324, 144, 264, 252, 288, 420, 340, 280, 336, 288, 260, 342, 294, 360, 408, 520, 360, 240, 496

Offset: 0

Author

N. J. A. Sloane, Feb 15 2014

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• I. J. Zucker, Exact Evaluation of Some New Lattice Sums, Symmetry, 2017, 9(12), 314.

Crossrefs

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^2), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

G.f.: 1 + 8*Sum{n >= 1} n*(q^n - q^(3*n) - q^(5*n) + q^(7*n))/(1 - q^(8*n)) - 2*Sum_{n >= 0} (-1)^((n^2+n)/2)*(2*n+1)q^(2*n+1)/(1 - q^(2*n+1)). See Zucker p. 5. Cf. A117000. - Peter Bala, Feb 25 2021