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

Author

Wolfdieter Lang, Apr 11 2003

Keywords

• nonn
• easy

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.

Examples

			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.
		

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.

Crossrefs

Cf. A082174 (number of primitive reduced forms).

Programs

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

Formula

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

Extensions

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