A249973 - OEIS

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

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OFFSET

1,4

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

LINKS

Table of n, a(n) for n=1..78.

EXAMPLE

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

PROG

(PARI) \\ see A249974

CROSSREFS

Cf. A249974, A006000.

Sequence in context: A341765 A104248 A358359 * A343854 A069349 A167404

Adjacent sequences: A249970 A249971 A249972 * A249974 A249975 A249976

KEYWORD

easy,nonn

AUTHOR

Kerry Mitchell, Nov 09 2014

EXTENSIONS

Edited. - Wolfdieter Lang, Jan 11 2015

STATUS

approved