Periodic coefficients and random Fibonacci sequences
type: Item
instance of: scholarly article
title: Periodic coefficients and random Fibonacci sequences
zbMATH Open document ID: 1295.11019
author: Karyn McLellan
published in: The Electronic Journal of Combinatorics
publication date: 2014-08-14T00:00:00Z
full work available at URL: v20i4p32
review text: Summary: The random Fibonacci sequence is defined by \(t_1 = t_2 = 1\) and \(t_n = \pm t_{n-1} + t_{n-2}\), for \(n \geq 3\), where each \(\pm\) sign is chosen at random with probability \(P(+) = P(-) = \frac{1}{2}\). Viswanath has shown that almost all random Fibonacci sequences grow exponentially at the rate \(1.13198824\ldots\). We will consider what happens to random Fibonacci sequences when we remove the randomness; specifically, we will choose coefficients which belong to the set \(\{1, -1\}\) and form periodic cycles. By rewriting our recurrences using matrix products, we will analyze sequence growth and develop criteria based on eigenvalue, trace and order for determining whether a given sequence is bounded, grows linearly or grows exponentially. Further, we will introduce an equivalence relation on the coefficient cycles such that each equivalence class has a common growth rate, and consider the number of such classes for a given cycle length.
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 05A05
Mathematics Subject Classification ID: 15B36
zbMATH DE Number: 6330363
zbMATH Keywords: random Fibonacci sequence
zbMATH Keywords: growth rate
zbMATH Keywords: periodic
zbMATH Keywords: matrix trace
describes a project that uses: OEIS
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6330363