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A113262 One quarter of the number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.

%I A113262 #26 Jan 06 2023 05:42:25
%S A113262 1,1,1,5,6,1,8,13,1,6,12,5,14,8,6,29,18,1,20,30,8,12,24,13,31,14,1,40,
%T A113262 30,6,32,61,12,18,48,5,38,20,14,78,42,8,44,60,6,24,48,29,57,31,18,70,
%U A113262 54,1,72,104,20,30,60,30,62,32,8,125,84,12,68,90,24,48,72,13,74,38,31
%N A113262 One quarter of the number of solutions to a^2 + b^2 + 3*c^2 + 3*d^2 = n.
%D A113262 B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 223, Entry 3(iv).
%D A113262 L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 229.
%D A113262 N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.3), p. 76, Eq. (31.43).
%H A113262 G. C. Greubel, <a href="/A113262/b113262.txt">Table of n, a(n) for n = 1..1000</a>
%F A113262 a(n) is multiplicative with a(3^e) = 1, a(2^e) = 2^(e+1) - 3, a(p^e) = (p^(e+1) - 1) / (p - 1) if p > 3.
%F A113262 G.f.: Sum_{k>0} k * x^k / (1 - (-x)^k) * Kronecker(9, k) = ((theta_3(x) * theta_3(x^3))^2 - 1) / 4.
%F A113262 A034896(n) = 4*a(n) if n > 0.
%F A113262 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = 0.411233... (A222171). - _Amiram Eldar_, Dec 01 2022
%F A113262 Dirichlet g.f.: zeta(s)*zeta(s-1)*(4^(1-s)-2^(1-s)+1)*(1-3^(1-s)). - _Amiram Eldar_, Jan 06 2023
%t A113262 A034896[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3])^2, {q, 0, n}]; Table[A034896[n]/4, {n, 1, 50}] (* _G. C. Greubel_, Dec 24 2017 *)
%t A113262 a[ n_] := If[ n < 1, 0, DivisorSum[ n, # KroneckerSymbol[ 9, #] (-1)^(n + #) &]]; (* _Michael Somos_, Nov 10 2018 *)
%o A113262 (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d * kronecker(9, d) * (-1)^(n-d)))};
%o A113262 (PARI) {a(n) = my(A, p, e); if(n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==3, 1, (p^(e+1) - 1) / (p - 1) - 2*(p==2))))};
%Y A113262 Cf. A034896, A222171.
%K A113262 nonn,mult
%O A113262 1,4
%A A113262 _Michael Somos_, Oct 21 2005