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A249973 Positive integers A when the positive roots of r^2 = Ar + B are listed in increasing order. 1

%I #20 Jan 12 2015 03:05:41

%S 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,

%T 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,

%U 5,2,4,1,3,2,5,1,4,3

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

%C 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.

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

%C 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.

%C For the A and B values leading to a positive integer limit r see a comment in A063929. - _Wolfdieter Lang_, Jan 12 2015

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

%o (PARI) \\ see A249974

%Y Cf. A249974, A006000.

%K easy,nonn

%O 1,4

%A _Kerry Mitchell_, Nov 09 2014

%E Edited. - _Wolfdieter Lang_, Jan 11 2015

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Last modified July 12 10:18 EDT 2024. Contains 374244 sequences. (Running on oeis4.)