cp's OEIS Frontend

A257003 Number of Zagier-reduced indefinite quadratic forms over the integers in two variables with discriminants D(n) = A079896(n).

%I A257003 #37 Jun 02 2025 06:47:09
%S A257003 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 A257003 21,17,8,15,18,14,18,21,13,12,20,27,11,16,26,18,17,25,23,21,13,20,25,
%U A257003 12,20,32,24,18,26,27,18,18,38,31,15,18,33
%N A257003 Number of Zagier-reduced indefinite quadratic forms over the integers in two variables with discriminants D(n) = A079896(n).
%C A257003 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 A257003 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 A257003 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 A257003 D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981.  See pages 122-123.
%H A257003 Robin Visser, <a href="/A257003/b257003.txt">Table of n, a(n) for n = 1..10000</a> (terms n = 1..1000 from Barry R. Smith).
%F A257003 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 A257003 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-(D-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 A257003 Table[Length[
%t A257003   Flatten[Select[
%t A257003     Table[{a, k}, {k,
%t A257003       Select[Range[Ceiling[-Sqrt[n]], Floor[Sqrt[n]]],
%t A257003        Mod[# - n, 2] == 0 &]}, {a,
%t A257003       Select[Divisors[(n - k^2)/4], # > (Sqrt[n] - k)/2 &]}],
%t A257003     UnsameQ[#, {}] &], 1]], {n,
%t A257003   Select[Range[
%t A257003     153], ! IntegerQ[Sqrt[#]] && (Mod[#, 4] == 0 ||
%t A257003        Mod[#, 4] == 1) &]}]
%o A257003 (SageMath)
%o A257003 def a(n):
%o A257003     i, D, ans = 1, Integer(5), 0
%o A257003     while(i < n):
%o A257003         D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
%o A257003     for k in range(-isqrt(D), isqrt(D)+1):
%o A257003         if ((D-k^2)%4 != 0): continue
%o A257003         for h in Integer((D-k^2)/4).divisors():
%o A257003             if h > (sqrt(D) - k)/2: ans += 1
%o A257003     return ans  # _Robin Visser_, Jun 01 2025
%Y A257003 Cf. A079896, A082175, A257004.
%K A257003 nonn
%O A257003 1,2
%A A257003 _Barry R. Smith_, Apr 14 2015
%E A257003 Offset corrected by _Robin Visser_, Jun 01 2025