0,1

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.

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

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

a(0)=2 because there are two reduced forms for D(0)=A079896(0)=5, namely [a,b,c]=[ -1, 1, 1] and [1, 1, -1]; here f(5)=3.

a(4)=6: for D(4)=A079896(4)=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(5)=4: for D(5)=A079896(5)=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(10)=6, D(10)=A079896(10)=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(10)=4.

Cf. A082174 (number of primitive reduced forms).

Adjacent sequences: A082172 A082173 A082174 this_sequence A082176 A082177 A082178

Sequence in context: A099735 A091279 A096002 this_sequence A129292 A126606 A077651

nonn,easy

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 11 2003