A monotonicity property for generalized Fibonacci sequences

Toufik Mansour, Mark Shattuck

Research output: Contribution to journalArticlepeer-review


Given k ≥ 2, let an be the sequence defined by the recurrence an = α1an-1 + ⋯ + αkan-k for n ≥ k, 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 annan-1n-1 n√an/n-1√an-1 is strictly decreasing for all n ≥ N, 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.

Original languageEnglish
Pages (from-to)585-592
Number of pages8
JournalMathematica Slovaca
Issue number3
StatePublished - 27 Jun 2017

Bibliographical note

Publisher Copyright:
© 2017 Mathematical Institute Slovak Academy of Sciences 2017.


  • k-Fibonacci numbers
  • log-concavity
  • monotonicity

ASJC Scopus subject areas

  • General Mathematics


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