%I #9 Jan 22 2017 16:33:37
%S 1,2,3,3,5,4,4,6,7,5,5,10,7,10,11,9,6,8,10,7,10,16,12,11,16,8,10,12,
%T 21,17,8,10,14,14,18,21,13,12,14,27,11,16,26,15,17,18,23,16,10,20,25,
%U 11,13,32,14,18,26,27,18,18,38,24,15,18,28
%N Number of primitive 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 A form is primitive if its coefficients are relatively prime.
%D D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.
%e For D=20, the a(5)=4 Zagier-reduced are x^2+6*x*y+4*y^2, 4*x^2+6*x*y+y^2 4*x^2+10*x*y+5*y^2, and 5*x^2+10*x*y+4*y^2.
%t Table[Length[
%t Select[Flatten[
%t 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],
%t GCD[#[[1]], #[[2]] +
%t 2*#[[1]], #[[1]] + #[[2]] - (n - #[[2]]^2)/(4*#[[1]])] == 1 &]], {n,
%t Select[Range[
%t 153], ! IntegerQ[Sqrt[#]] && (Mod[#, 4] == 0 || Mod[#, 4] == 1) &]}]
%Y Cf. A079896, A082174, A257003.
%K nonn
%O 0,2
%A _Barry R. Smith_, Apr 17 2015