

A307359


Class number a(n) of indefinite binary quadratic forms with discriminant 4*A000037(n) for n >= 1.


3



1, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 4, 1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 1, 4, 2, 2, 2, 4, 4, 3, 2, 4, 4, 1, 4, 2, 2, 2, 1, 2, 4, 2, 4, 2, 1, 2, 4, 4, 2, 2, 2, 4, 1, 2, 4, 2, 4, 2, 2, 2, 4, 2, 4, 1, 2, 4, 2, 2, 4
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

This is a subsequence of A087048, See the formula.
This sequence is relevant for the Pell forms [1, 0,  D(n)], with D(n) = A000037(n) and discriminant 4*D(n).
The Buell reference, Table 2B, pp. 241243, gives only the class numbers, called there H, for A000037(n) squarefree and not congruent to 1 modulo 4. E.g., a(3), related to discriminant 4*5 = 20, is not treated there; also a(6) for discriminant 32 = 4*(2*2^2) does not appear there.
For the a(n) cycles of primitive reduced forms of discriminant 4*A000037(n) see the W. lang link in A324251, Table 2 and Table 1, for n = 1..30.  Wolfdieter Lang, Apr 19 2019


REFERENCES

D. A. Buell, Binary Quadratic Forms, Springer, 1989.


LINKS

Table of n, a(n) for n=1..70.


FORMULA

a(n) gives the number of distinct cycles of primitive reduced forms of discriminant 4*A000037(n).
a(n) = A087048(e(n)), with e(n) the position of the nth even term of A079896, for n >= 1.


EXAMPLE

a(1) = 1 because 4*A000037(1) = 4*2 = 8 = A079896(e(1)) with e(1) = 1 and A087048(1) = 1.
a(12) = 4 because the twelfth even number of A079896 is 60 at position e(12) = 22, and A087048(22) = 4.
The cycle for discriminant 8 is [[1, 2, 1], [1, 2, 1]].
The four 2cycles for discriminant 60 are [[1, 6, 6], [6, 6, 1]], [[1, 6, 6], [6, 6, 1]], [[2, 6, 3], [3, 6, 2]] and [[2, 6, 3], [3, 6, 2]].


CROSSREFS

Cf. A000037, A079896, A082174, A087048.
Sequence in context: A297774 A043533 A043558 * A297775 A043534 A043559
Adjacent sequences: A307356 A307357 A307358 * A307360 A307361 A307362


KEYWORD

nonn


AUTHOR

Wolfdieter Lang, Apr 04 2019


STATUS

approved



