OFFSET
0,2
COMMENTS
A binary quadratic form A*x^2 + B*x*y + C*y^2 with integer coefficients A, B, and C and positive discriminant D = B^2 - 4*A*C is Zagier-reduced if A>0, C>0, and B>A+C. (This differs from the classical reduced forms defined by Lagrange.) There are finitely many Zagier-reduced forms of given discriminant.
Zagier defines a reduction operation on binary quadratic forms with positive discriminants, which permutes the reduced forms. The reduced forms are thereby partitioned into disjoint cycles.
REFERENCES
D. B. Zagier, Zetafunktionen und quadratische Korper, Springer, 1981.
FORMULA
With D=n^2-4, a(n) equals the number of pairs (a,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), a > (sqrt(D) - k)/2, a exactly dividing (D-k^2)/4.
EXAMPLE
For n=3, the a(3) = 3 forms in the principal cycle of discriminant A079896(3) = 13 are x^2 + 5*x*y + 3*y^2, 3*x^2 + 5*x*y + y^2, and 3*x^2 + 7*x*y + 3*y^2.
CROSSREFS
KEYWORD
nonn
AUTHOR
Barry R. Smith, Apr 16 2015
STATUS
approved