A236923 - OEIS

%I #24 Mar 04 2023 15:09:29

%S 1,6,12,8,8,36,48,16,24,78,72,24,32,84,96,48,24,108,156,40,48,192,144,

%T 48,96,186,168,80,64,180,288,64,24,288,216,96,104,228,240,112,144,252,

%U 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

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

%H Seiichi Manyama, <a href="/A236923/b236923.txt">Table of n, a(n) for n = 0..10000</a>

%H Olivia X. M. Yao and Ernest X. W. Xia, <a href="https://doi.org/10.1016/j.disc.2013.11.011">Combinatorial proofs of five formulas of Liouville</a>, Discrete Math. 318 (2014), 1--9. MR3141622.

%F See Maple code.

%F G.f.: theta_3(q)^3*theta_3(q^4), where theta_3() is the Jacobi theta function. - _Ilya Gutkovskiy_, Aug 01 2018

%p with(numtheory);

%p s:=n-> if whattype(n) = integer then sigma(n) else 0; fi;

%p f:=proc(n) global s;

%p   if (n mod 4) = 0 then 8*s(n/4)-32*s(n/16)

%p elif (n mod 4) = 2 then 12*s(n/2)

%p elif (n mod 4) = 3 then 2*s(n)

%p else 6*s(n);

%p fi; end;

%p [seq(f(n),n=1..100)];

%p # a(0)=1 must be added separately

%t EllipticTheta[3, 0, q]^3*EllipticTheta[3, 0, q^4] + O[q]^80 // CoefficientList[#, q]& (* _Jean-François Alcover_, Mar 04 2023, after _Ilya Gutkovskiy_ *)

%Y Cf. A097057, A236924.

%Y 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.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Feb 14 2014