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Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 8*d^2 = n.
8

%I #15 Oct 30 2018 03:36:07

%S 1,2,2,6,6,4,12,4,4,18,4,20,30,12,36,24,10,32,14,24,48,32,36,40,24,18,

%T 28,34,36,60,60,28,28,40,16,56,78,44,108,68,8,72,24,40,144,60,72,112,

%U 30,46,42,64,84,116,120,40,72,84,28,116,96,60,180,68,34,120,60,64,192,80

%N Number of integer solutions to a^2 + 2*b^2 + 3*c^2 + 8*d^2 = n.

%C a(n) > 0 for n >= 0.

%H Seiichi Manyama, <a href="/A320189/b320189.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^3) * theta_3(q^8).

%t CoefficientList[Series[Product[EllipticTheta[3, 0, q^k], {k, 1, 3}]*EllipticTheta[3,0,q^8], {q, 0, 80}], q] (* _G. C. Greubel_, Oct 29 2018 *)

%o (PARI) q='q+O('q^80); Vec(prod(k=1,3, eta(q^(2*k))^5/(eta(q^k)* eta(q^(4*k)))^2 )*eta(q^(16))^5/(eta(q^8)* eta(q^(32)))^2 ) \\ _G. C. Greubel_, Oct 29 2018

%Y Cf. A320138, A320139, A320140, A033712, A320188, A320190, A320191.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Oct 07 2018