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A082175 Number of reduced indefinite quadratic forms over the integers in two variables with discriminants D(n)=A079896(n). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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
FORMULA
a(n)= number of reduced indefinite quadratic forms over the integers for D(n)=A079896(n) (counting also nonprimitive forms).
EXAMPLE
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.
CROSSREFS
Cf. A082174 (number of primitive reduced forms).
Sequence in context: A322352 A091279 A096002 * A129292 A126606 A285699
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Apr 11 2003
STATUS
approved

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