%I #30 Feb 15 2018 16:23:46
%S 1,2,3,3,5,5,4,6,7,5,7,10,7,10,11,9,7,11,13,7,10,16,12,11,16,13,10,14,
%T 21,17,8,15,18,14,18,21,13,12,20,27,11,16,26,18,17,25,23,21,13,20,25,
%U 12,20,32,24,18,26,27,18,18,38,31,15,18,33
%N Number of Zagier-reduced indefinite quadratic forms over the integers in two variables with discriminants D(n) = A079896(n).
%C An indefinite quadratic form in two variables over the integers, A*x^2 + B*x*y + C*y^2 with discriminant D = B^2 - 4*A*C > 0, 0 or 1 (mod 4) and not a square, is called Zagier-reduced if A>0, C>0, and B>A+C.
%C This definition is from Zagier's 1981 book, and differs from the older and more common notion of reduced form due to Lagrange (see A082175 for this definition).
%C The number of pairs of integers (h,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), h > (sqrt(D) - k)/2, h exactly dividing (D-k^2)/4, where D=D(n) is the discriminant being considered.
%D D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
%H Barry R. Smith, <a href="/A257003/b257003.txt">Table of n, a(n) for n = 0..999</a>
%F a(n) equals the number of pairs (h,k) with |k| < sqrt(D), k^2 congruent to D (mod 4), h > (sqrt(D) - k)/2, h exactly dividing (D-k^2)/4, where D=D(n) is the discriminant being considered.
%e For D=20, the pairs (h,k) as above are: (1,4), (2,2), (4,2), (5,0), (4,-2). From these, the a(6)=5 Zagier-reduced forms may be enumerated as h*x^2 + (k+2*h)*x*y + (k+h-(n-k^2)/4*h)*y^2, yielding x^2+6*x*y+4*y^2, 2*x^2+6*x*y+2*y^2, 4*x^2+10*x*y+5*y^2, 5*x^2+10*x*y+4*y^2, and 4*x^2+6*x*y+y^2.
%t Table[Length[
%t Flatten[Select[
%t Table[{a, k}, {k,
%t Select[Range[Ceiling[-Sqrt[n]], Floor[Sqrt[n]]],
%t Mod[# - n, 2] == 0 &]}, {a,
%t Select[Divisors[(n - k^2)/4], # > (Sqrt[n] - k)/2 &]}],
%t UnsameQ[#, {}] &], 1]], {n,
%t Select[Range[
%t 153], ! IntegerQ[Sqrt[#]] && (Mod[#, 4] == 0 ||
%t Mod[#, 4] == 1) &]}]
%Y Cf. A079896, A082175, A257004.
%K nonn
%O 0,2
%A _Barry R. Smith_, Apr 14 2015