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%I #24 Jul 12 2017 15:17:01
%S -1,0,0,1,1,0,0,1,1,0,0,1,4,0,0,2,3,0,0,1,2,0,0,3,2,0,0,4,2,0,0,3,3,0,
%T 0,2,5,0,0,4,2,0,0,1,4,0,0,5,10,0,0,2,2,0,0,4,4,0,0,3,4,0,0,5,7,0,0,1,
%U 4,0,0,7,3,0,0,7,4,0,0,5,6,0,0,3,4,0,0,6,2,0,0,2,6,0,0,8,6,0,0,3
%N Numerator 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).
%C H(n) is usually called the Hurwitz class number.
%C a(n) = 0 if n = 1 or 2 (mod 4).
%D D. Zagier, The Eichler-Selberg Trace Formula on SL_2(Z), Appendix to S. Lang, Introduction to Modular Forms, Springer, 1976.
%H Charles R Greathouse IV, <a href="/A058305/b058305.txt">Table of n, a(n) for n = 0..10000</a>
%H N. Lygeros, O. Rozier, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Lygeros/lygeros5.html">A new solution to the equation tau(rho) == 0 (mod p)</a>, J. Int. Seq. 13 (2010) # 10.7.4.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Hurwitz_class_number">Hurwitz class number</a>.
%F H(n) = A259825(n) / 12. - _Michael Somos_, Jul 05 2015
%e -1/12, 0, 0, 1/3, 1/2, 0, 0, 1, 1, ...
%t 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 // Numerator (* _Jean-François Alcover_, Apr 02 2017, after _Michael Somos_ *)
%o (PARI) H(n)=sumdiv(core(n,1)[2],d,my(D=-n/d^2);if(D%4<2,qfbclassno(D)/max(1,D+6)))
%o a(n)=if(n,numerator(H(n)),-1) \\ _Charles R Greathouse IV_, Apr 25 2013
%o (PARI) {a(n) = numerator( qfbhclassno( n))}; /* _Michael Somos_, Jul 06 2015 */
%Y Cf. A058306, A259825.
%K sign,frac
%O 0,13
%A _N. J. A. Sloane_, Dec 09 2000
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