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Search: id:A004531
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| A004531 |
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Number of integer solutions to x^2+4y^2=n. |
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+0
5
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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, 4, 0, 0, 0, 4, 4, 0, 0, 8, 4, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 8, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 8, 0, 0, 8, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 0, 8, 2, 0, 0, 0, 8, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 12, 4, 0, 0, 8
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OFFSET
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0,2
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REFERENCES
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J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.
B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373 Entry 32.
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FORMULA
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Coefficients in expansion of Sum_{ i, j = -inf .. inf } q^(i^2+4*j^2).
a(4n+2)=a(4n+3)=0.
G.f. Sum{k>=0} a(k)q^(k/2) = (theta2^2+theta3^2+theta4^2)/2.
Expansion of (eta(q^2)*eta(q^8))^5/(eta(q)^2*eta(q^4)^4*et(q^16)^2) in powers of q.
G.f.: phi(x)*phi(x^4) = phi(x^4)^2+2*x*psi(x^4)^2 where phi(x), psi(x) are Ramanujan theta functions.
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PROGRAM
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(PARI) a(n)=if(n<1, n==0, 2*qfrep([1, 0; 0, 4], n)[n]) /* Michael Somos Jul 04 2005 */
(PARI) {a(n)=local(A, e1, e2, e4); if(n<0, 0, A=x*O(x^n); e1=eta(x^2+A); e2=eta(x^4+A); e4=eta(x^8+A); polcoeff( (e2^12+e1^8*e4^4+4*x*e1^4*e4^8)/(2*e1^4*e2^2*e4^4), n))}
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CROSSREFS
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A004018(n)=a(4n), A004020(n)=a(4n+1).
Adjacent sequences: A004528 A004529 A004530 this_sequence A004532 A004533 A004534
Sequence in context: A108885 A072740 A080964 this_sequence A134014 A072071 A045836
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KEYWORD
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nonn
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AUTHOR
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njas
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