A236922 Number of integer solutions to a^2 + b^2 + 4*c^2 + 4*d^2 = n.

1, 4, 4, 0, 8, 24, 16, 0, 24, 52, 24, 0, 32, 56, 32, 0, 24, 72, 52, 0, 48, 128, 48, 0, 96, 124, 56, 0, 64, 120, 96, 0, 24, 192, 72, 0, 104, 152, 80, 0, 144, 168, 128, 0, 96, 312, 96, 0, 96, 228, 124, 0, 112, 216, 160, 0, 192, 320, 120, 0, 192, 248, 128, 0, 24, 336, 192, 0, 144, 384, 192, 0, 312, 296, 152, 0, 160, 384, 224, 0, 144, 484, 168, 0, 256, 432, 176, 0

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, Ernest X. W. Xia, Combinatorial proofs of five formulas of Liouville, Discrete Math. 318 (2014), 1--9. MR3141622.

Crossrefs

Cf. A097057, A236923.

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 4*s(n/2)
elif (n mod 4) = 3 then 0
else 4*s(n); fi; end;
[seq(f(n),n=1..100)];
# a(0)=1 must be added separately

Mathematica

s[n_] := If[IntegerQ[n], DivisorSigma[1, n], 0]; a[n_] := Which[Mod[n, 4] == 0 , 8*s[n/4]-32*s[n/16], Mod[n, 4] == 2, 4*s[n/2], Mod[n, 4] == 3, 0, True, 4*s[n]]; a[0] = 1; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 06 2014, after Maple *)

Formula

See Maple code.

G.f.: theta_3(q)^2*theta_3(q^4)^2, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 03 2018