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A096936
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Half of number of integer solutions to the equation x^2+3y^2=n.
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6
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1, 0, 1, 3, 0, 0, 2, 0, 1, 0, 0, 3, 2, 0, 0, 3, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 1, 6, 0, 0, 2, 0, 0, 0, 0, 3, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 3, 3, 0, 0, 6, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 6, 0, 0, 2, 0, 1, 0, 0, 6, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 0, 3, 0, 0, 2, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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REFERENCES
| N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.25).
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LINKS
| M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211
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FORMULA
| 2*a(n) = A033716(n), if n>0.
Multiplicative with a(2^e)=3(1+(-1)^e)/2, a(3^e)=1, a(p^e)=(1+(-1)^e) if p=2 (mod 3) and a(p^e)=1+e if p=1 (mod 3).
G.f.: ((Sum_{k} x^(k^2))(Sum_{k} x^(3k^2))-1)/2.
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MAPLE
| sigmamr := proc(n, m, r) local a, d ; a := 0 ; for d in numtheory[divisors](n) do if modp(d, m) = r then a := a+1 ; end if; end do: a; end proc:
A002324 := proc(n) sigmamr(n, 3, 1)-sigmamr(n, 3, 2) ; end proc:
A096936 := proc(n) A002324(n) +2*(sigmamr(n, 12, 4)-sigmamr(n, 12, 8) ); end proc:
seq(A096936(n), n=1..90) ; # R. J. Mathar, Mar 23 2011
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PROG
| (PARI) a(n)=if(n<1, 0, 1/2*polcoeff(sum(k=1, sqrtint(n), 2*x^k^2, 1+x^n*O(x))*sum(k=1, sqrtint(n\3), 2*x^(3*k^2), 1+x^n*O(x)), n))
(PARI) a(n)=if(n<1, 0, qfrep([1, 0; 0, 3], n)[n]) /* Michael Somos Jun 05 2005 */
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CROSSREFS
| Sequence in context: A176788 A193291 * A115979 A067168 A099475 A120569
Adjacent sequences: A096933 A096934 A096935 * A096937 A096938 A096939
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KEYWORD
| nonn,mult
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AUTHOR
| Michael Somos, Jul 15 2004
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