A257161 The length of the period under Zagier-reduction of the principal indefinite quadratic binary form of discriminant D(n) = A079896(n).

1, 2, 1, 3, 5, 4, 1, 2, 2, 5, 1, 4, 7, 6, 11, 3, 1, 2, 10, 7, 2, 7, 1, 11, 9, 8, 2, 4, 21, 7, 1, 2, 4, 9, 6, 21, 2, 3, 1, 27, 11, 10, 3, 5, 17, 6, 23, 16, 1, 2, 8, 11, 2, 15, 2, 6, 2, 27, 1

Offset: 1

Author

Barry R. Smith, Apr 16 2015

Keywords

• nonn

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.

There is a unique Zagier-reduced form with A=1 for each discriminant in A079896. The cycle containing this form is the principal cycle. a(n) is the length of this cycle for the discriminant D=A079896(n).

Examples

			For n=4, the a(4) = 3 forms in the principal cycle of discriminant A079896(4) = 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.
		

References

• D. B. Zagier, Zetafunktionen und quadratische Korper, Springer, 1981.

Crossrefs

Cf. A226166.

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.

Extensions

Offset corrected by Robin Visser, Jun 08 2025