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A058306
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).
3
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
OFFSET
0,1
COMMENTS
H(n) is usually called the Hurwitz class number.
a(n) = 1 unless n is of the form 3k^2 or 4k^2. - Charles R Greathouse IV, Apr 25 2013
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.
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 // Denominator (* 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, denominator(H(n)), 12) \\ Charles R Greathouse IV, Apr 25 2013
(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 */
CROSSREFS
Sequence in context: A334731 A335948 A010209 * A010207 A010206 A174096
KEYWORD
nonn,frac,easy
AUTHOR
N. J. A. Sloane, Dec 09 2000
STATUS
approved