A236924 Number of integer solutions to a^2 + 2*b^2 + 2*c^2 + 4*d^2 = n.

1, 2, 4, 8, 8, 12, 16, 16, 24, 26, 24, 24, 32, 28, 32, 48, 24, 36, 52, 40, 48, 64, 48, 48, 96, 62, 56, 80, 64, 60, 96, 64, 24, 96, 72, 96, 104, 76, 80, 112, 144, 84, 128, 88, 96, 156, 96, 96, 96, 114, 124, 144, 112, 108, 160, 144, 192, 160, 120, 120, 192, 124, 128, 208, 24, 168, 192, 136, 144, 192, 192

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. A000118, A320149, A320150, A320151, A320152, A320153.

Programs

Maple

with(LinearAlgebra);
L:=proc(a,b,c,d,M) local n,hit,ul,vl,wl,xl,u,v,w,x,su,sv,sw,sx;
hit:=Array(0..M); ul:=floor(sqrt(M/a)); vl:=floor(sqrt(M/b));
wl:=floor(sqrt(M/c)); xl:=floor(sqrt(M/d));
for u from -ul to ul do
su:=a*u^2;
for v from -vl to vl do
sv:=su+b*v^2;
for w from -wl to wl do
sw:=sv+c*w^2;
for x from -xl to xl do
sx:=sw+d*x^2;
if sx <= M then hit[sx]:=hit[sx]+1; fi;
od; od; od; od;
[seq(hit[n],n=0..M)];
end;
L(1,2,2,4,60);

Mathematica

CoefficientList[EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^2]^2*EllipticTheta[3, 0, q^4] + O[q]^100, q] (* Jean-François Alcover, Feb 15 2023, after Ilya Gutkovskiy *)

Formula

Yao and Xia give an explicit formula for a(n) due to Liouville.

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