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

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, 21, 17, 8, 15, 18, 14, 18, 21, 13, 12, 20, 27, 11, 16, 26, 18, 17, 25, 23, 21, 13, 20, 25, 12, 20, 32, 24, 18, 26, 27, 18, 18, 38, 31, 15, 18, 33

Offset: 1

Author

Barry R. Smith, Apr 14 2015

Keywords

• nonn

Comments

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.

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).

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.

Examples

			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.
		

References

• D. B. Zagier, Zetafunktionen und quadratische Körper, Springer, 1981. See pages 122-123.

Links

• Robin Visser, Table of n, a(n) for n = 1..10000 (terms n = 1..1000 from Barry R. Smith).

Crossrefs

Cf. A079896, A082175, A257004.

Programs

Mathematica

Table[Length[
  Flatten[Select[
    Table[{a, k}, {k,
      Select[Range[Ceiling[-Sqrt[n]], Floor[Sqrt[n]]],
       Mod[# - n, 2] == 0 &]}, {a,
      Select[Divisors[(n - k^2)/4], # > (Sqrt[n] - k)/2 &]}],
    UnsameQ[#, {}] &], 1]], {n,
  Select[Range[
    153], ! IntegerQ[Sqrt[#]] && (Mod[#, 4] == 0 ||
       Mod[#, 4] == 1) &]}]

SageMath

def a(n):
    i, D, ans = 1, Integer(5), 0
    while(i < n):
        D += 1; i += 1*(((D%4) in [0, 1]) and (not D.is_square()))
    for k in range(-isqrt(D), isqrt(D)+1):
        if ((D-k^2)%4 != 0): continue
        for h in Integer((D-k^2)/4).divisors():
            if h > (sqrt(D) - k)/2: ans += 1
    return ans  # Robin Visser, Jun 01 2025

Formula

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.

Extensions

Offset corrected by Robin Visser, Jun 01 2025