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

%I A236923 #24 Mar 04 2023 15:09:29
%S A236923 1,6,12,8,8,36,48,16,24,78,72,24,32,84,96,48,24,108,156,40,48,192,144,
%T A236923 48,96,186,168,80,64,180,288,64,24,288,216,96,104,228,240,112,144,252,
%U A236923 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 A236923 Number of integer solutions to a^2 + b^2 + c^2 + 4*d^2 = n.
%H A236923 Seiichi Manyama, <a href="/A236923/b236923.txt">Table of n, a(n) for n = 0..10000</a>
%H A236923 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 A236923 See Maple code.
%F A236923 G.f.: theta_3(q)^3*theta_3(q^4), where theta_3() is the Jacobi theta function. - _Ilya Gutkovskiy_, Aug 01 2018
%p A236923 with(numtheory);
%p A236923 s:=n-> if whattype(n) = integer then sigma(n) else 0; fi;
%p A236923 f:=proc(n) global s;
%p A236923   if (n mod 4) = 0 then 8*s(n/4)-32*s(n/16)
%p A236923 elif (n mod 4) = 2 then 12*s(n/2)
%p A236923 elif (n mod 4) = 3 then 2*s(n)
%p A236923 else 6*s(n);
%p A236923 fi; end;
%p A236923 [seq(f(n),n=1..100)];
%p A236923 # a(0)=1 must be added separately
%t A236923 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 A236923 Cf. A097057, A236924.
%Y A236923 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 A236923 nonn
%O A236923 0,2
%A A236923 _N. J. A. Sloane_, Feb 14 2014