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A087048
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Class numbers of indefinite quadratic forms over the integers in two variables with discriminant D = D(n) = A079896(n), n>=0.
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3
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1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 4, 1, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 4, 1, 1, 2, 4, 2, 1, 2, 1, 1, 2, 4, 2, 1, 2, 2, 2, 2, 4, 1, 4, 2, 4, 3, 1, 2, 2, 4, 1, 4, 2, 1, 4, 4, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 4, 1, 1, 2, 2, 4, 4, 2, 2, 1, 2, 2, 2, 4, 4, 4, 2, 3, 2, 1, 2, 2, 4
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| An indefinite quadratic form over the integers in two variables F(x,y) := a*x^2 + b*x*y + c*y^2 has discriminant D := b^2 - 4*a*c >0 not a square (a and c non-vanishing); that is D=D(n)= A079896(n) = [5,8,12,13,17,20,21,...], n>=0.
For a given discriminant D from A079896(n) a reduced form [a,b,c] is defined by b>0 and f(D)-min(|2*a|,|2*c|) =< b < f(D), with f(D) := ceiling(sqrt(D)).
For a given discriminant D from A079896(n) every primitive reduced form [a,b,c] defines a periodic chain of such forms by applying repeatedly the transformation R(t)*[a,b,c]=[a'(t),b'(t),c'(t)]=[c,-b+2*c*t,F(1,t)] with uniquely defined t= ceiling((f(D)+b)/(2*c))-1 if c>0 and t=-(ceiling((f(D)+b)/(2*|c|)-1)) if c<0. The number of such different) periodic chains of primitive reduced forms is called the class number for this (indefinite) discriminant D from A079896(n).
A primitive form [a,b,c] has gcd(a,b,c)=1.
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REFERENCES
| A. Scholz and B. Schoeneberg, Einfuehrung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, ch. 31, pp. 112 ff.
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LINKS
| S. R. Finch, Class number theory
W. Lang, Table for n=0,...,135.
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EXAMPLE
| n=2, D(2) = A079896(2) = 12, a(2) = 2 because there are the following two periodic chains of primitive reduced forms [a,b,c] (both with period length 2): [[ -2, 2, 1], [1, 2, -2]] and [[ -1, 2, 2], [2, 2, -1]].
n=13, D(13) = A079896(13) = 40, a(13) = 2 because there are the following two periodic chains of primitive reduced forms [a,b,c] (with period length 6 resp. 2): [[ -3, 2, 3], [3, 4, -2], [ -2, 4, 3], [3, 2, -3], [ -3, 4, 2], [2, 4, -3]] and [[ -1, 6, 1], [1, 6, -1]].
n=35, D(35) = A079896(35) = 89, a(35) = 1 because there is only one periodic chain of primitive reduced forms [a,b,c] (with period length 14): [[ -5, 3, 4], [4, 5, -4], [ -4, 3, 5], [5, 7, -2], [ -2, 9, 1], [1, 9, -2], [ -2, 7, 5], [5, 3, -4], [ -4, 5, 4], [4, 3, -5], [ -5, 7, 2], [2, 9, -1], [ -1, 9, 2], [2, 7, -5]]. See p. 116 of the Scholz/Schoeneberg reference which starts with the form [1, 9, -2].
n=62, D(62) = A079896(62) = 148, a(62) = 3 because there are three periodic chains of primitive reduced forms [a,b,c] (with period length 6 and 6 and 2, resp.): [[ -7, 6, 4], [4, 10, -3], [ -3, 8, 7], [7, 6, -4], [ -4, 10, 3], [3, 8, -7]] and [[ -4, 6, 7], [7, 8, -3], [ -3, 10, 4], [4, 6, -7], [ -7, 8, 3], [3, 10, -4]] and [[ -1, 12, 1], [1, 12, -1]]. See p. 116 of the Scholz/Schoeneberg reference which starts with the forms [4, 10, -3] and [3, 10, -4] and [1, 12, -1], resp.
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CROSSREFS
| See A006375 for another version. Cf. A079896.
Sequence in context: A122586 A079487 A069010 * A109700 A087742 A072530
Adjacent sequences: A087045 A087046 A087047 * A087049 A087050 A087051
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KEYWORD
| nonn
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Aug 07 2003
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