

A087048


Class numbers of indefinite quadratic forms over the integers in two variables with discriminant D = D(n) = A079896(n), n>=0.


13



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
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OFFSET

0,3


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 nonvanishing); 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).  Wolfdieter Lang, Jun 07 2013
A primitive form [a,b,c] has gcd(a,b,c)=1.
See the Appendix 2 of the Buell reference. pp. 235243, for the class numbers, called H(D), for the fundamental discriminants 0 < D < 10000. Table 2A gives the class numbers for squarefree D == 1 (mod 4) and Table 2B the ones for D == 0 (mod 4), with D/4 squarefree and not congruent to 1 modulo 4 (compare Buell, p. 69, 1. and 2.).  Wolfdieter Lang, May 29 2013
For an online program for D < 10^6 see the Keith Matthews link.  Wolfdieter Lang, Jul 24 2019


REFERENCES

D. A. Buell, Binary Quadratic Forms, Springer, 1989.
A. Scholz and B. Schoeneberg, Einführung in die Zahlentheorie, 5. Aufl., de Gruyter, Berlin, New York, 1973, ch. 31, pp. 112 ff.


LINKS

Table of n, a(n) for n=0..104.
S. R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Keith Matthews, Finding the class number h(d) of primitive binary quadratic forms of positive discriminant d Wolfdieter Lang, Table for n=0, ..., 135


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.


CROSSREFS

See A006374 for another version. Cf. A079896.
Sequence in context: A353332 A353362 A256122 * A109700 A087742 A340996
Adjacent sequences: A087045 A087046 A087047 * A087049 A087050 A087051


KEYWORD

nonn


AUTHOR

Wolfdieter Lang, Aug 07 2003


STATUS

approved



