A236923 Number of integer solutions to a^2 + b^2 + c^2 + 4*d^2 = n.

1, 6, 12, 8, 8, 36, 48, 16, 24, 78, 72, 24, 32, 84, 96, 48, 24, 108, 156, 40, 48, 192, 144, 48, 96, 186, 168, 80, 64, 180, 288, 64, 24, 288, 216, 96, 104, 228, 240, 112, 144, 252, 384, 88, 96, 468, 288, 96, 96, 342, 372, 144, 112, 324, 480, 144, 192, 480, 360, 120, 192, 372, 384, 208, 24, 504, 576, 136, 144, 576, 576, 144, 312, 444, 456, 248, 160, 576, 672

Offset: 0

Author

N. J. A. Sloane, Feb 14 2014

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Olivia X. M. Yao and Ernest X. W. Xia, Combinatorial proofs of five formulas of Liouville, Discrete Math. 318 (2014), 1--9. MR3141622.

Crossrefs

Cf. A097057, 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

with(numtheory);
s:=n-> if whattype(n) = integer then sigma(n) else 0; fi;
f:=proc(n) global s;
  if (n mod 4) = 0 then 8*s(n/4)-32*s(n/16)
elif (n mod 4) = 2 then 12*s(n/2)
elif (n mod 4) = 3 then 2*s(n)
else 6*s(n);
fi; end;
[seq(f(n),n=1..100)];
# a(0)=1 must be added separately

Mathematica

EllipticTheta[3, 0, q]^3*EllipticTheta[3, 0, q^4] + O[q]^80 // CoefficientList[#, q]& (* Jean-François Alcover, Mar 04 2023, after Ilya Gutkovskiy *)

Formula

See Maple code.

G.f.: theta_3(q)^3*theta_3(q^4), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018