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A329290 Number of ordered triples (i, j, k) of integers such that n = i^2 + 4*j^2 + 4*k^2.

%I A329290 #8 Feb 16 2025 08:33:58
%S A329290 1,2,0,0,6,8,0,0,12,10,0,0,8,8,0,0,6,16,0,0,24,16,0,0,24,10,0,0,0,24,
%T A329290 0,0,12,16,0,0,30,8,0,0,24,32,0,0,24,24,0,0,8,18,0,0,24,24,0,0,48,16,
%U A329290 0,0,0,24,0,0,6,32,0,0,48,32,0,0,36,16,0,0,24,32
%N A329290 Number of ordered triples (i, j, k) of integers such that n = i^2 + 4*j^2 + 4*k^2.
%C A329290 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A329290 G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(5/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A169783.
%H A329290 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A329290 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A329290 Euler transform of period 16 sequence [2, -3, 2, 3, 2, -3, 2, -7, 2, -3, 2, 3, 2, -3, 2, -3, ...].
%F A329290 Expansion of phi(x) * phi(x^4)^2 = phi(x^4)^3 + 2*x*phi(x^4)*psi(x^4)^2 in powers of x where phi(), psi() are Ramanujan theta functions.
%F A329290 G.f.: theta_3(q) * theta_3(q^4)^2, where theta_3() is the Jacobi theta function.
%F A329290 a(4*n) = A005875(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = a(4*n + 3) = 0.
%e A329290 G.f. = 1 + 2*x + 6*x^4 + 8*x^5 + 12*x^8 + 10*x^9 + 8*x^12 + 8*x^13 + ...
%t A329290 a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, x] EllipticTheta[3, 0, x^4]^2, {x, 0, n}];
%o A329290 (PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A)^10 / (eta(x + A)^2 * eta(x^4 + A)^6 * eta(x^16 + A)^4), n))};
%o A329290 (Magma) A := Basis( ModularForms( Gamma0(16), 3/2), 77); A[1] + 2*A[2];
%Y A329290 Cf. A005875, A045834, A169783.
%K A329290 nonn
%O A329290 0,2
%A A329290 _Michael Somos_, Nov 17 2019