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A072068 Number of integer solutions to the equation 2x^2+y^2+8z^2=m for an odd number m=2n-1.

%I A072068 #35 Jul 02 2023 06:59:06
%S A072068 2,4,0,0,10,12,0,0,16,12,0,0,10,16,0,0,16,24,0,0,32,12,0,0,18,24,0,0,
%T A072068 16,36,0,0,32,12,0,0,16,28,0,0,34,36,0,0,48,24,0,0,16,36,0,0,32,36,0,
%U A072068 0,32,24,0,0,26,24,0,0
%N A072068 Number of integer solutions to the equation 2x^2+y^2+8z^2=m for an odd number m=2n-1.
%C A072068 Related to primitive congruent numbers A006991.
%C A072068 Assuming the Birch and Swinnerton-Dyer conjecture, the odd number 2n-1 is a congruent number if it is squarefree and a(n) = 2*A072069(n).
%C A072068 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A072068 T. D. Noe, <a href="/A072068/b072068.txt">Table of n, a(n) for n=1..10000</a>
%H A072068 Clay Mathematics Institute, <a href="https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/">The Birch and Swinnerton-Dyer Conjecture</a>
%H A072068 Department of Pure Maths., Univ. Sheffield, <a href="https://web.archive.org/web/20040206183520/http://www.shef.ac.uk/~puremath/theorems/congruent.html">Pythagorean triples and the congruent number problem</a>
%H A072068 Karl Rubin, <a href="https://web.archive.org/web/20110511082917/http://math.Stanford.EDU/~rubin/lectures/sumo/">Elliptic curves and right triangles</a>
%H A072068 J. B. Tunnell, <a href="http://dx.doi.org/10.1007/BF01389327">A classical Diophantine problem and modular forms of weight 3/2</a>, Invent. Math., 72 (1983), 323-334.
%F A072068 Expansion of 2 * x * phi(x) * psi(x^4) * phi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Jun 08 2012
%F A072068 Expansion of 2 * q^(1/2) * eta(q^2)^5 * eta(q^8)^7 / (eta(q)^2 * eta(q^4)^5 * eta(q^16)^2) in powers of q. - _Michael Somos_, Feb 19 2015
%e A072068 a(2) = 4 because (1,1,0), (-1,1,0), (1,-1,0) and (-1,-1,0) are solutions when m=3.
%e A072068 G.f. = 2*x + 4*x^2 + 10*x^5 + 12*x^6 + 16*x^9 + 12*x^10 + 10*x^13 + 16*x^14 + 16*x^17 + ...
%e A072068 G.f. = 2*q + 4*q^3 + 10*q^9 + 12*q^11 + 16*q^17 + 12*q^19 + 10*q^25 + 16*q^27 + ...
%t A072068 maxN=128; soln1=Table[0, {maxN/2}]; xMax=Ceiling[Sqrt[maxN/2]]; yMax=Ceiling[Sqrt[maxN]]; zMax=Ceiling[Sqrt[maxN/8]]; Do[n=2x^2+y^2+8z^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]; soln1[[(n+1)/2]]+=s], {x, 0, xMax}, {y, 0, yMax}, {z, 0, zMax}]
%t A072068 (* Second program: *)
%t A072068 phi[x_] := EllipticTheta[3, 0, x];
%t A072068 psi[x_] := (1/2)*x^(-1/8)*EllipticTheta[2, 0, x^(1/2)];
%t A072068 s = Series[2*x*phi[x]*psi[x^4]*phi[x^4], {x, 0, 100}];
%t A072068 A072068 = CoefficientList[s, x] // Rest (* _Jean-François Alcover_, Feb 16 2015, after _Michael Somos_ *)
%o A072068 (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)^7 / (eta(x + A)^2 * eta(x^4 + A)^5 * eta(x^16 + A)^2), n))}; /* _Michael Somos_, Dec 26 2019 */
%Y A072068 Cf. A006991, A003273, A072069, A072070, A072071, A080917.
%K A072068 nonn
%O A072068 1,1
%A A072068 _T. D. Noe_, Jun 13 2002