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A246928 Number of integer solutions to x^2 + 3*y^2 + 3*z^2 = n.

%I A246928 #20 Feb 16 2025 08:33:23
%S A246928 1,2,0,4,10,0,4,16,0,2,8,0,12,8,0,16,26,0,0,24,0,8,8,0,20,10,0,4,32,0,
%T A246928 8,48,0,8,16,0,10,8,0,32,40,0,8,24,0,0,16,0,28,18,0,24,40,0,4,64,0,8,
%U A246928 8,0,32,24,0,16,58,0,16,24,0,16,16,0,0,16,0,28
%N A246928 Number of integer solutions to x^2 + 3*y^2 + 3*z^2 = n.
%C A246928 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A246928 G. C. Greubel, <a href="/A246928/b246928.txt">Table of n, a(n) for n = 0..1000</a>
%H A246928 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A246928 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A246928 Expansion of phi(q) * phi(q^3)^2 in powers of q where phi() is a Ramanujan theta function.
%F A246928 Expansion of eta(q^2)^5 * eta(q^6)^10 / (eta(q)^2 * eta(q^3)^4 * eta(q^4)^2 * eta(q^12)^4) in powers of q.
%F A246928 Euler transform of period 12 sequence [ 2, -3, 6, -1, 2, -9, 2, -1, 6, -3, 2, -3, ...].
%F A246928 G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 24^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A034933.
%F A246928 G.f. theta_3(q) * theta_3(q^3)^2.
%F A246928 a(3*n) = A034933(n). a(3*n + 1) = 2 * A246926(n). a(3*n + 2) = 0.
%e A246928 G.f. = 1 + 2*q + 4*q^3 + 10*q^4 + 4*q^6 + 16*q^7 + 2*q^9 + 8*q^10 + ...
%t A246928 a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3]^2, {q, 0, n}]; (* _Michael Somos_, Jan 08 2015 *)
%o A246928 (PARI) {a(n) = if( n<1, n==0, qfrep( [ 1, 0, 0; 0, 3, 0; 0, 0, 3], n)[n] * 2)};
%o A246928 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^6 + A)^10 / (eta(x + A)^2 * eta(x^3 + A)^4 * eta(x^4 + A)^2 * eta(x^12 + A)^4), n))};
%o A246928 (Magma) A := Basis( ModularForms( Gamma1(12), 3/2), 76); A[1] + 2*A[2] + 4*A[4] + 10*A[5];
%Y A246928 Cf. A034933, A246926.
%K A246928 nonn
%O A246928 0,2
%A A246928 _Michael Somos_, Sep 07 2014