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A monotonicity property for generalized Fibonacci sequences

Toufik Mansour and Mark Shattuck
From the journal Mathematica Slovaca


Given k ≥ 2, let an be the sequence defined by the recurrence an = α1an–1 + … + αkank for nk, with initial values a0 = a1 = … = ak–2 = 0 and ak–1 = 1. We show under a couple of assumptions concerning the constants αi that the ratio annan1n1 is strictly decreasing for all nN, for some N depending on the sequence, and has limit 1. In particular, this holds in the cases when all of the αi are unity or when all of the αi are zero except for the first and last, which are unity. Furthermore, when k = 3 or k = 4, it is shown that one may take N to be an integer less than 12 in each of these cases.

(Communicated by Federico Pellarin)


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Received: 2014-10-19
Accepted: 2015-10-11
Published Online: 2017-6-7
Published in Print: 2017-6-27

© 2017 Mathematical Institute Slovak Academy of Sciences