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A monotonicity property for generalized Fibonacci sequences

  • Toufik Mansour EMAIL logo and Mark Shattuck
Published/Copyright: June 7, 2017
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Mathematica Slovaca
From the journal Mathematica Slovaca Volume 67 Issue 3

Abstract

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 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.

MSC 2010: Primary 11B39; Secondary 11B39; 11B75
Keywords: monotonicity; log-concavity; k-Fibonacci numbers

(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

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https://doi.org/10.1515/ms-2016-0292
Keywords for this article
monotonicity; log-concavity; k-Fibonacci numbers

Articles in the same Issue

  2. Some results in local Hilbert algebras
  4. A note on residuated po-groupoids and lattices with antitone involutions
  6. The first law of cubology for the Rubik’s Revenge
  8. Conditional associativity in orthomodular lattices
  10. A monotonicity property for generalized Fibonacci sequences
  12. Systems of conditionally independent sets
  14. On the k-th order lfsr sequence with public key cryptosystems
  16. On parametric spaces of bicentric quadrilaterals
  18. Character sums with an explicit evaluation
  20. Some remarks on formal power series and formal Laurent series
  22. The co-rank of the fundamental group: The direct product, the first Betti number, and the topology of foliations
  24. On prime and semiprime generalized hyperaction of hypermonoid
  26. On O’Malley ϱ-upper continuous functions
  28. Uniqueness theorems for finitely additive probabilities on quantum structures
  30. Some results on differential-difference analogues of Brück conjecture
  32. Oscillation of neutral second order half-linear differential equations without commutativity in deviating arguments
  34. On support, separation and decomposition theorems for t-Wright-concave functions
  36. A characterization of holomorphic bivariate functions of bounded index
  38. Asymptotic expansion of double Laplace-type integrals with a curve of minimal points and application to an exit time problem
  40. The geometry of two-valued subsets of Lp-spaces
  42. Hemi-slant submanifolds as warped products in a nearly Kaehler manifold
  44. A comparison of two tests in multivariate models
  46. Modelling prescription behaviour of general practitioners
  48. Characterization of congruence lattices of principal p-algebras
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