A082175 Number of reduced indefinite quadratic forms over the integers in two variables with discriminants D(n)=A079896(n).

2, 2, 4, 2, 6, 4, 4, 4, 8, 2, 6, 8, 6, 8, 10, 4, 6, 8, 12, 2, 8, 12, 8, 6, 12, 8, 8, 6, 18, 12, 4, 8, 16, 8, 12, 14, 8, 4, 16, 18, 6, 8, 20, 8, 14, 16, 14, 12, 6, 12, 16, 4, 14, 20, 16, 8, 20, 14, 8, 8, 28, 20, 10, 4, 22, 16, 10, 20, 20, 16, 12, 8, 20, 2, 20, 24, 20, 10, 24, 16, 8, 12

Offset

1,1

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 reduced if b>0 and f(D) - min(|2*a|,|2*c|) <= b < f(D), with f(D) := ceiling(sqrt(D)). See the Scholz-Schoeneberg reference for this definitions.

References

A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, ch.IV, par.31, p. 112.

Links

Robin Visser, Table of n, a(n) for n = 1..10000

Henri Cohen, A Course in Computational Algebraic Number Theory, Grad. Texts in Math., Vol. 138, Springer-Verlag, Berlin, 1993. xii+534 pp. See Definition 5.6.2 on page 257.

Formula

a(n)= number of reduced indefinite quadratic forms over the integers for D(n)=A079896(n) (counting also nonprimitive forms).

Example

a(1)=2 because there are two reduced forms for D(1)=A079896(1)=5, namely [a,b,c]=[-1, 1, 1] and [1, 1, -1]; here f(5)=3.
a(5)=6: for D(5)=A079896(5)=17 (f(17)=5) the 6 reduced [a,b,c] forms are [[-2, 1, 2], [2, 1, -2], [-2, 3, 1], [-1, 3, 2], [1, 3, -2], [2, 3, -1]]. They are all primitive (that is a,b and c are relatively prime).
a(6)=4: for D(6)=A079896(6)=20 (f(20)=5) there are four reduced forms: [-2, 2, 2], [2, 2, -2], [-1, 4, 1] and [1, 4, -1], Here two of them are nonprimitive, namely [-2, 2, 2], [2, 2, -2].
a(11)=6, D(11)=A079896(11)=32 (f(32)=6); the 6 reduced forms are [-4, 4, 1], [-2, 4, 2], [-1, 4, 4], [1, 4, -4], [2, 4, -2] and [4, 4, -1]. Two of them are nonprimitive, namely [ -2, 4, 2] and [2, 4, -2]. Therefore A082174(11)=4.

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 b in range(1, isqrt(D)+1):
        if ((D-b^2)%4 != 0): continue
        for a in Integer((D-b^2)/4).divisors():
            if abs(sqrt(D) - 2*a) < b: ans += 1
    return 2*ans  # Robin Visser, May 31 2025

Crossrefs

Cf. A082174 (number of primitive reduced forms).

Sequence in context: A322352 A091279 A096002 * A129292 A126606 A285699

Adjacent sequences: A082172 A082173 A082174 * A082176 A082177 A082178

Keyword

nonn,easy

Author

Wolfdieter Lang, Apr 11 2003

Extensions

Offset corrected and more terms from Robin Visser, May 31 2025

Status

approved