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%I #16 Dec 26 2019 13:41:24
%S 2,4,0,0,6,4,0,0,4,4,0,0,2,8,0,0,12,8,0,0,16,12,0,0,10,16,0,0,12,20,0,
%T 0,16,4,0,0,12,12,0,0,14,20,0,0,20,8,0,0,4,20,0,0,8,12,0,0,24,8,0,0,
%U 14,8,0,0
%N Number of integer solutions to the equation 2x^2+y^2+32z^2=m for an odd number m=2n-1.
%C Related to primitive congruent numbers A006991.
%C Assuming the Birch and Swinnerton-Dyer conjecture, the odd number 2n-1 is a congruent number if it is squarefree and 2 a(n) = A072068(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="/A072069/b072069.txt">Table of n, a(n) for n=1..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>
%F Expansion of 2 * x * phi(x) * psi(x^4) * phi(x^16) in powers of x where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Jun 08 2012
%F Expansion of 2 * q^(1/2) * eta(q^2)^5 * eta(q^8)^2 * eta(q^32)^5 / (eta(q)^2 * eta(q^4)^3 * eta(q^16)^2 * eta(q^64)^2) in powers of q. - _Michael Somos_, Dec 26 2019
%e a(2) = 4 because (1,1,0), (-1,1,0), (1,-1,0) and (-1,-1,0) are solutions when m=3.
%e G.f. = 2*x + 4*x^2 + 6*x^5 + 4*x^6 + 4*x^9 + 4*x^10 + 2*x^13 + 8*x^14 + ... - _Michael Somos_, Dec 26 2019
%e G.f. = 2*q + 4*q^3 + 6*q^9 + 4*q^11 + 4*q^17 + 4*q^19 + 2*q^25 + 8*q^27 + 12*q^33
%e + ...
%t maxN=128; soln2=Table[0, {maxN/2}]; xMax=Ceiling[Sqrt[maxN/2]]; yMax=Ceiling[Sqrt[maxN]]; zMax=Ceiling[Sqrt[maxN/32]]; Do[n=2x^2+y^2+32z^2; If[OddQ[n]&&n<maxN, s=8; If[x==0, s=s/2]; If[y==0, s=s/2]; If[z==0, s=s/2]; soln2[[(n+1)/2]]+=s], {x, 0, xMax}, {y, 0, yMax}, {z, 0, zMax}]
%o (PARI) {a(n) = my(A); n--; if( n<0, 0, A = x * O(x^n); polcoeff( 2 * eta(x^2 + A)^5 * eta(x^8 + A)^2 * eta(x^32 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^16 + A)^2 * eta(x^64 + A)^2), n))}; /* _Michael Somos_, Dec 26 2019 */
%Y Cf. A006991, A003273, A072068, A072070, A072071, A080918.
%K nonn
%O 1,1
%A _T. D. Noe_, Jun 13 2002
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