A058305 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).

-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, 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, 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

Offset

0,13

Comments

H(n) is usually called the Hurwitz class number.

a(n) = 0 if n = 1 or 2 (mod 4).

References

D. Zagier, The Eichler-Selberg Trace Formula on SL_2(Z), Appendix to S. Lang, Introduction to Modular Forms, Springer, 1976.

Links

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000

N. Lygeros, O. Rozier, A new solution to the equation tau(rho) == 0 (mod p), J. Int. Seq. 13 (2010) # 10.7.4.

Wikipedia, Hurwitz class number.

Formula

H(n) = A259825(n) / 12. - Michael Somos, Jul 05 2015

Example

-1/12, 0, 0, 1/3, 1/2, 0, 0, 1, 1, ...

Mathematica

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 *)

Prog

(PARI) H(n)=sumdiv(core(n, 1)[2], d, my(D=-n/d^2); if(D%4<2, qfbclassno(D)/max(1, D+6)))
a(n)=if(n, numerator(H(n)), -1) \\ Charles R Greathouse IV, Apr 25 2013
(PARI) {a(n) = numerator( qfbhclassno( n))}; /* Michael Somos, Jul 06 2015 */

Crossrefs

Cf. A058306, A259825.

Sequence in context: A317448 A292900 A177893 * A193717 A020808 A198364

Adjacent sequences: A058302 A058303 A058304 * A058306 A058307 A058308

Keyword

sign,frac

Author

N. J. A. Sloane, Dec 09 2000

Status

approved