A257161 - OEIS

%I #7 Apr 19 2015 01:01:16

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

%C 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 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 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 D. B. Zagier, Zetafunktionen und quadratische Korper, Springer, 1981.

%F 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 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.

%Y Cf. A226166.

%K nonn

%O 0,2

%A _Barry R. Smith_, Apr 16 2015