A249973 Positive integers A when the positive roots of r^2 = Ar + B are listed in increasing order.

1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 3, 1, 2, 1, 3, 2, 1, 3, 2, 1, 3, 2, 1, 1, 2, 3, 4, 1, 2, 3, 1, 4, 2, 1, 3, 2, 4, 1, 3, 2, 1, 4, 3, 2, 1, 4, 3, 2, 1, 1, 2, 3, 4, 5, 1, 2, 3, 4, 1, 2, 5, 3, 1, 4, 2, 1, 3, 5, 2, 4, 1, 3, 2, 5, 1, 4, 3

Offset: 1

Author

Kerry Mitchell, Nov 09 2014

Keywords

• easy
• nonn

Comments

Generalize the Fibonacci sequence recurrence equation as: F_(n+1) = A*F_n + B*F_(n-1), where A and B are positive integers. As n goes to infinity, the ratio F_n / F_(n-1) approaches the positive real number r = (A + sqrt(A*A + 4B))/2. This sequence gives the A values in increasing order of r.

In case of a tie in r values, then sort in increasing order of sqrt(A*A + B*B).

This A sequence appears to be the ordinal transform of the B sequence (A249974) and vice versa. The associative arrays of A and B are transposes. The first row of A's associative array seems to be A006000.

For the A and B values leading to a positive integer limit r see a comment in A063929. - Wolfdieter Lang, Jan 12 2015

Examples

			a(6) = 2 because the 6th smallest value of r (approximately 2.732050808) is that for A=2, B=2.
		

Crossrefs

Cf. A249974, A006000.

Programs

PARI

\\ see A249974

Extensions

Edited. - Wolfdieter Lang, Jan 11 2015