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Search: id:A113262
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| A113262 |
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One quarter of the number of solutions to a^2+b^2+3*c^2+3*d^2=n. |
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+0
2
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| 1, 1, 1, 5, 6, 1, 8, 13, 1, 6, 12, 5, 14, 8, 6, 29, 18, 1, 20, 30, 8, 12, 24, 13, 31, 14, 1, 40, 30, 6, 32, 61, 12, 18, 48, 5, 38, 20, 14, 78, 42, 8, 44, 60, 6, 24, 48, 29, 57, 31, 18, 70, 54, 1, 72, 104, 20, 30, 60, 30, 62, 32, 8, 125, 84, 12, 68, 90, 24, 48, 72, 13, 74, 38, 31
(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 III, Springer-Verlag, see p. 223 Entry 3(iv).
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 229.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.3), p. 76, Eq. (31.43).
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FORMULA
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a(n) is multiplicative and a(3^e) = 1, a(2^e) = 2^(e+1)-3, a(p^e) = (p^(e+1)-1)/(p-1) if p>3.
G.f.: Sum_{k>0} k x^k/(1-(-x)^k) kronecker(9, k) = ((theta_3(x)theta_3(x^3))^2-1)/4.
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PROGRAM
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(PARI) a(n)=if(n<1, n==0, sumdiv(n, d, d^2*kronecker(-3, d)*(-1)^(n-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==3, 1, (p^(e+1)-1)/(p-1)-2*(p==2)))))}
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CROSSREFS
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Cf. A034896(n)=4 a(n) if n>0.
Sequence in context: A080130 A020798 A021182 this_sequence A131947 A105577 A054655
Adjacent sequences: A113259 A113260 A113261 this_sequence A113263 A113264 A113265
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KEYWORD
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nonn,mult
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AUTHOR
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Michael Somos, Oct 21 2005
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