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%I #26 Sep 08 2022 08:44:51
%S 1,2,2,4,4,4,8,8,6,6,8,4,8,12,0,8,12,8,10,12,8,8,24,8,8,14,8,16,16,4,
%T 0,16,6,16,16,8,12,20,24,8,24,8,16,20,8,20,0,16,24,18,10,8,24,12,32,
%U 24,0,16,24,12,16,20,0,24,12,8,16,28,16,16,48,8,30,32,8,20,24,16,0,16,24,18
%N Number of integer solutions to the equation x^2 + 2*y^2 + 4*z^2 = n.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.
%H G. C. Greubel, <a href="/A033717/b033717.txt">Table of n, a(n) for n = 0..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of phi(q) * phi(q^2) * phi(q^4) in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Sep 03 2014
%F Euler transform of period 16 sequence [2, -1, 2, -2, 2, -1, 2, -5, 2, -1, 2, -2, 2, -1, 2, -3, ...]. - _Michael Somos_, Sep 03 2014
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - _Michael Somos_, Sep 03 2014
%F a(2*n + 1) = 2 * A045828(n). a(4*n) = A014455(n). a(4*n + 1) = 2 * A213625(n). a(4*n + 2) = 2 * A246811(n). a(4*n + 3) = 4 * A213624(n). - _Michael Somos_, Sep 03 2014
%F a(8*n) = A005875(n). a(8*n + 1) = 2 * A213622(n). a(8*n + 2) = 2 * A045834(n). a(8*n + 7) = 8 * A033763(n). - _Michael Somos_, Sep 03 2014
%F a(16*n) = A004015(n). a(16*n + 2) = 2 * A213022(n). a(16*n + 6) = 8 *
%F A008443(n). a(16*n + 8) = 2 * A045826(n). a(16*n + 10) = 8 * A045831(n). a(16*n + 14) = 0. - _Michael Somos_, Sep 03 2014
%F G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4).
%e G.f. = 1 + 2*q + 2*q^2 + 4*q^3 + 4*q^4 + 4*q^5 + 8*q^6 + 8*q^7 + 6*q^8 + ...
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^4], {q, 0, n}]; (* _Michael Somos_, Sep 03 2014 *)
%o (PARI) {a(n) = my(G); if( n<0, 0, G = [1, 0, 0; 0, 2, 0; 0, 0, 4]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n)), n))}; /* _Michael Somos_, Sep 03 2014 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^8 + A)^3 / (eta(x + A)^2 * eta(x^16 + A)^2), n))}; /* _Michael Somos_, Sep 03 2014 */
%o (Magma) A := Basis( ModularForms( Gamma1(16), 3/2), 82); A[1] + 2*A[2] + 2*A[3] + 4*A[4] + 4*A[5] + 4*A[6] + 8*A[7] + 8*A[8] + 6*A[9] + 8*A[10] + 4*A[11]; /* _Michael Somos_, Sep 03 2014 */
%Y Cf. A004015, A008443, A014455, A033763, A045826, A045828, A045831, A045834, A213022, A213622, A213624, A213625, A246811.
%K nonn
%O 0,2
%A _N. J. A. Sloane_
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