cp's OEIS Frontend

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

%I A257161 #10 Jun 09 2025 00:53:43
%S A257161 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,
%T A257161 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
%N A257161 The length of the period under Zagier-reduction of the principal indefinite quadratic binary form of discriminant D(n) = A079896(n).
%C A257161 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.
%C A257161 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.
%C A257161 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).
%D A257161 D. B. Zagier, Zetafunktionen und quadratische Korper, Springer, 1981.
%F A257161 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.
%e A257161 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.
%Y A257161 Cf. A226166.
%K A257161 nonn
%O A257161 1,2
%A A257161 _Barry R. Smith_, Apr 16 2015
%E A257161 Offset corrected by _Robin Visser_, Jun 08 2025