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%I #21 Mar 12 2021 22:24:42
%S 1,2,0,0,4,4,0,0,6,6,0,0,8,12,0,0,12,8,0,0,8,8,0,0,8,14,0,0,16,4,0,0,
%T 6,16,0,0,12,20,0,0,24,8,0,0,8,20,0,0,24,18,0,0,24,12,0,0,0,16,0,0,16,
%U 20,0,0,12,8,0,0,16,16,0,0,30,32,0,0,24,16,0,0,24,18,0,0,16,24,0,0,24,16
%N Number of integer solutions to the equation 4*x^2 + y^2 + 8*z^2 = n.
%C Related to primitive congruent numbers A006991.
%C Assuming the Birch and Swinnerton-Dyer conjecture, the even number 2n is a congruent number if it is squarefree and a(n) = 2 A072071(n).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
%H T. D. Noe, <a href="/A072070/b072070.txt">Table of n, a(n) for n = 0..10000</a>
%H Clay Mathematics Institute, <a href="http://www.claymath.org/prizeproblems/birchsd.htm">The Birch and Swinnerton-Dyer Conjecture</a>
%H Department of Pure Maths., Univ. Sheffield, <a href="http://www.shef.ac.uk/~puremath/theorems/congruent.html">Pythagorean triples and the congruent number problem</a>
%H Karl Rubin, <a href="http://math.Stanford.EDU/~rubin/lectures/sumo/">Elliptic curves and right triangles</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^4) * phi(q^8) in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Jun 09 2012
%F Euler transform of period 32 sequence [2, -3, 2, 1, 2, -3, 2, -2, 2, -3, 2, 1, 2, -3, 2, -5, 2, -3, 2, 1, 2, -3, 2, -2, 2, -3, 2, 1, 2, -3, 2, -3, ...]. - _Michael Somos_, Feb 11 2003
%F a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A014455(n). - _Michael Somos_, Jun 08 2012
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(7/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A080917. - _Michael Somos_, Jul 23 2018
%e a(4) = 4 because (1, 0, 0), (-1, 0, 0), (0, 2, 0) and (0, -2, 0) are solutions.
%e G.f. = 1 + 2*q + 4*q^4 + 4*q^5 + 6*q^8 + 6*q^9 + 8*q^12 + 12*q^13 + 12*q^16 + 8*q^17 + ...
%t maxN=128; soln3=Table[0, {maxN/2}]; xMax=Ceiling[Sqrt[maxN/8]]; yMax=Ceiling[Sqrt[maxN/2]]; zMax=Ceiling[Sqrt[maxN/16]]; Do[n=4x^2+y^2+8z^2; If[n>0&&n<=maxN/2, s=8; If[x==0, s=s/2]; If[y==0, s=s/2]; If[z==0, s=s/2]; soln3[[n]]+=s], {x, 0, xMax}, {y, 0, yMax}, {z, 0, zMax}]
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^4] EllipticTheta[ 3, 0, q^8], {q, 0, n}]; (* _Michael Somos_, Jul 23 2018 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^5 * eta(x^4 + A)^-4 * eta(x^8 + A)^3 * eta(x^16 + A)^3 * eta(x^32 + A)^-2, n))}; /* _Michael Somos_, Feb 11 2003 */
%Y Cf. A006991, A003273, A072068, A072069, A072071, A080917.
%K nonn
%O 0,2
%A _T. D. Noe_, Jun 13 2002
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