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Search: id:A113406
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| A113406 |
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Half the number of integer solutions to x^2+4y^2=n. |
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+0
3
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| 1, 0, 0, 2, 2, 0, 0, 2, 1, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 4, 0, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 4, 2, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 4, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 6, 2, 0, 0, 4, 0
(list; graph; listen)
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OFFSET
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1,4
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REFERENCES
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B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 373 Entry 32.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.
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FORMULA
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a(n) is multiplicative with a(2) = 0, a(2^e) = 2 if e>1, a(p^e) = e+1 if p == 1 (mod 4), a(p^e) = (1+(-1)^e)/2 if p == 3 (mod 4)
G.f.: (theta_3(q)theta_3(q^4)-1)/2.
a(4n+2)=a(4n+3)=0.
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PROGRAM
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(PARI) a(n)=if(n<1, 0, qfrep([1, 0; 0, 4], n)[n])
(PARI) {a(n)=if(n<1, 0, if(n%4==1, sumdiv(n, d, (-1)^(d\2)), if(n%4==0, 2*sumdiv(n, d, kronecker(-4, d)))))}
(PARI) {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, 2*(e>1), if(p%4==3, (1+(-1)^e)/2, e+1)))))}
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CROSSREFS
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A004531(n)=2 a(n) if n>0. A008441(n)=a(4n+1). A004018(n)=2*a(4n) if n>0.
Sequence in context: A125226 A059080 A062070 this_sequence A134015 A151851 A033461
Adjacent sequences: A113403 A113404 A113405 this_sequence A113407 A113408 A113409
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KEYWORD
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nonn,mult
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AUTHOR
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Michael Somos, Oct 28 2005
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