A307359 Class number a(n) of indefinite binary quadratic forms with discriminant 4*A000037(n) for n >= 1.

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

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. 241-243, 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 n-th 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 2-cycles 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 * A368860 A297775 A043534

Adjacent sequences: A307356 A307357 A307358 * A307360 A307361 A307362

Keyword

nonn

Author

Wolfdieter Lang, Apr 04 2019

Status

approved