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A072071 Number of integer solutions to the equation 4x^2+y^2+32z^2=n.

%I A072071 #21 Feb 16 2025 08:32:46
%S A072071 1,2,0,0,4,4,0,0,4,2,0,0,0,4,0,0,4,4,0,0,8,0,0,0,0,6,0,0,0,4,0,0,6,4,
%T A072071 0,0,12,12,0,0,16,8,0,0,0,12,0,0,8,10,0,0,24,4,0,0,0,12,0,0,0,12,0,0,
%U A072071 12,8,0,0,16,8,0,0,20,12,0,0,0,8,0,0,8,6,0,0,16,16,0,0,0,4,0,0,0,8,0,0,8
%N A072071 Number of integer solutions to the equation 4x^2+y^2+32z^2=n.
%C A072071 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A072071 Related to primitive congruent numbers A006991.
%C A072071 Assuming the Birch and Swinnerton-Dyer conjecture, the even number 2n is a congruent number if it is squarefree and 2 a(n) = A072070(n).
%D A072071 J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
%H A072071 T. D. Noe, <a href="/A072071/b072071.txt">Table of n, a(n) for n = 0..10000</a>
%H A072071 Clay Mathematics Institute, <a href="http://www.claymath.org/prizeproblems/birchsd.htm">The Birch and Swinnerton-Dyer Conjecture</a>
%H A072071 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 A072071 Karl Rubin, <a href="http://math.Stanford.EDU/~rubin/lectures/sumo/">Elliptic curves and right triangles</a>
%H A072071 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A072071 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F A072071 Expansion of phi(x) * phi(x^4) * phi(x^32) in powers of x where phi() is a Ramanujan theta function.
%F A072071 a(4*n + 2) = a(4*n + 3) = 0. - _Michael Somos_, Jun 08 2012
%e A072071 a(4) = 4 because (1,0,0), (-1,0,0), (0,2,0) and (0,-2,0) are solutions.
%e A072071 1 + 2*x + 4*x^4 + 4*x^5 + 4*x^8 + 2*x^9 + 4*x^13 + 4*x^16 + 4*x^17 + 8*x^20 + ...
%t A072071 J12[q_] := Sum[q^n^2, {n, -10, 10}]; CoefficientList[Series[J12[q]J12[q^4]J12[q^32], {q, 0, 100}], q]
%o A072071 (PARI) {a(n) = local(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)^5 * eta(x^16 + A)^-2 * eta(x^32 + A)^-2 * eta(x^64 + A)^5 * eta(x^128 + A)^-2, n))}
%Y A072071 Cf. A006991, A003273, A072068, A072069, A072070.
%K A072071 nonn
%O A072071 0,2
%A A072071 _T. D. Noe_, Jun 13 2002
%E A072071 More terms from _Vladeta Jovovic_, Jun 16 2002