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%I #19 Oct 06 2018 09:29:40
%S 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,
%T 124,56,0,64,120,96,0,24,192,72,0,104,152,80,0,144,168,128,0,96,312,
%U 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
%N Number of integer solutions to a^2 + b^2 + 4*c^2 + 4*d^2 = n.
%H Seiichi Manyama, <a href="/A236922/b236922.txt">Table of n, a(n) for n = 0..10000</a>
%H Olivia X. M. Yao, 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)^2*theta_3(q^4)^2, where theta_3() is the Jacobi theta function. - _Ilya Gutkovskiy_, Aug 03 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 4*s(n/2)
%p elif (n mod 4) = 3 then 0
%p else 4*s(n); fi; end;
%p [seq(f(n),n=1..100)];
%p # a(0)=1 must be added separately
%t 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 *)
%Y Cf. A097057, A236923.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Feb 14 2014
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