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

1,2

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.

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).

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.

Example

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.

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) &]}]

Prog

(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

Crossrefs

Cf. A079896, A082175, A257004.

Sequence in context: A273493 A193404 A072923 * A131922 A260718 A064631

Adjacent sequences: A257000 A257001 A257002 * A257004 A257005 A257006

Keyword

nonn

Author

Barry R. Smith, Apr 14 2015

Extensions

Offset corrected by Robin Visser, Jun 01 2025

Status

approved