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A072071
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Number of integer solutions to the equation 4x^2+y^2+32z^2=n.
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9
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A10054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (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).
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REFERENCES
| J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..10000
M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
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
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FORMULA
| G.f.: phi(x)*phi(x^4)*phi(x^32) where phi(x)=1+2x+2x^4+2x^9+... is a Ramanujan theta function.
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EXAMPLE
| a(4) = 4 because (1,0,0), (-1,0,0), (0,2,0) and (0,-2,0) are solutions.
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MATHEMATICA
| J12[q_] := Sum[q^n^2, {n, -10, 10}]; CoefficientList[Series[J12[q]J12[q^4]J12[q^32], {q, 0, 100}], q]
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PROG
| (PARI) a(n)=local(X); if(n<0, 0, X=x+x*O(x^n); polcoeff(eta(X)^-2*eta(X^2)^5*eta(X^4)^-4*eta(X^8)^5*eta(X^16)^-2*eta(X^32)^-2*eta(X^64)^5*eta(X^128)^-2, n))
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CROSSREFS
| Cf. A006991, A003273, A072068, A072069, A072070.
Sequence in context: A080964 A004531 A134014 * A045836 A072070 A137828
Adjacent sequences: A072068 A072069 A072070 * A072072 A072073 A072074
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KEYWORD
| nonn
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AUTHOR
| T. D. Noe (noe(AT)sspectra.com), Jun 13 2002
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EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 16 2002
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