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A058306
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Denominator of H(n), where H(0)=-1/12, H(n) = number of equivalence classes of positive definite quadratic forms a*x^2+b*x*y+c*y^2 with discriminant b^2-4ac = -n, counting forms equivalent to x^2+y^2 (resp. x^2+x*y+y^2) with multiplicity 1/2 (resp. 1/3).
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3
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12, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
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OFFSET
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0,1
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COMMENTS
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H(n) is usually called the Hurwitz class number.
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REFERENCES
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D. Zagier, The Eichler-Selberg Trace Formula on SL_2(Z), Appendix to S. Lang, Introduction to Modular Forms, Springer, 1976.
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LINKS
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FORMULA
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EXAMPLE
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-1/12, 0, 0, 1/3, 1/2, 0, 0, 1, 1, ...
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MATHEMATICA
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terms = 100; gf[m_] := With[{r = Range[-m, m]}, -2 Sum[(-1)^k*x^(k^2 + k)/(1 + (-x)^k)^2, {k, r}]/EllipticTheta[3, 0, x] - 2 Sum[(-1)^k*x^(k^2 + 2 k)/(1 + x^(2k))^2, {k, r}]/EllipticTheta[3, 0, -x]]; CoefficientList[ gf[terms // Sqrt // Ceiling] + O[x]^terms, x]/12 // Denominator (* Jean-François Alcover, Apr 02 2017, after Michael Somos *)
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PROG
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(PARI) H(n)=sumdiv(core(n, 1)[2], d, my(D=-n/d^2); if(D%4<2, qfbclassno(D)/max(1, D+6)))
(PARI) a(n)=if(n, my(D=4-valuation(n, 3)%2); denominator(if(issquare(n/D) && n%D==0, qfbclassno(-D)/max(1, 6-D))), 12) \\ Charles R Greathouse IV, Apr 25 2013
(PARI) {a(n) = denominator( qfbhclassno( n))}; /* Michael Somos, Jul 06 2015 */
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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