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 A072071 Number of integer solutions to the equation 4x^2+y^2+32z^2=n. 9
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700). Related to primitive congruent numbers A006991. 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). REFERENCES J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334. LINKS T. D. Noe, Table of n, a(n) for n = 0..10000 Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture Department of Pure Maths., Univ. Sheffield, Pythagorean triples and the congruent number problem Karl Rubin, Elliptic curves and right triangles Eric Weisstein's World of Mathematics, Ramanujan Theta Functions FORMULA Expansion of phi(x) * phi(x^4) * phi(x^32) in powers of x where phi() is a Ramanujan theta function. a(4*n + 2) = a(4*n + 3) = 0. - Michael Somos, Jun 08 2012 EXAMPLE a(4) = 4 because (1,0,0), (-1,0,0), (0,2,0) and (0,-2,0) are solutions. 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 + ... MATHEMATICA J12[q_] := Sum[q^n^2, {n, -10, 10}]; CoefficientList[Series[J12[q]J12[q^4]J12[q^32], {q, 0, 100}], q] PROG (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))} CROSSREFS Cf. A006991, A003273, A072068, A072069, A072070. Sequence in context: A080964 A134014 A004531 * A329264 A045836 A182056 Adjacent sequences:  A072068 A072069 A072070 * A072072 A072073 A072074 KEYWORD nonn AUTHOR T. D. Noe, Jun 13 2002 EXTENSIONS More terms from Vladeta Jovovic, Jun 16 2002 STATUS approved

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Last modified February 24 01:11 EST 2020. Contains 332195 sequences. (Running on oeis4.)