Random Fibonacci sequences and the number $1.13198824\dots$

Abstract

For the familiar Fibonacci sequence (defined by f 1 = f 2 = 1, and fn = f n−1 + f n−2 for n > 2), fn increases exponentially with n at a rate given by the golden ratio (1 + √ 5)/2 = 1.61803398. .. . But for a simple modification with both additions and subtractions-the random Fibonacci sequences defined by t 1 = t 2 = 1, and for n > 2, tn = ±t n−1 ± t n−2 , where each ± sign is independent and either + or − with probability 1/2-it is not even obvious if |tn| should increase with n. Our main result is that n |tn| → 1.13198824. .. as n → ∞ with probability 1. Finding the number 1.13198824. .. involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal measure, a computer calculation, and a rounding error analysis to validate the computer calculation.

How do random Fibonacci sequences grow?

“…This definition is equivalent to ours only in the case p = 1/2, which explains why the graph of the Lyapunov exponent drawn on Fig. 5 in [11] is different from our Fig. 2.…”

“…In the case p = 1/2, (|F n |) and ( F n ) have the same distribution law as the sequence (|t n |) studied by Viswanath [11]. In his paper, using Furstenberg's formula [4] (see also [1,Chap.…”

Random geometric series

“…While traditional integer sequences are deterministic, there is a growing interest in stochastic counterparts of fundamental sequences and their relevance to disordered or random systems. For example, the random Fibonacci sequence x n = x n−1 ± x n−2 [2,3,4] has links with various topics in condensed matter physics, dynamical systems, products of random matrices, etc. (see e.g.…”

A novel approach for assessing the susceptibility of Escherichia coli to antibiotics

“…However, the occurrence of cell division for bacteria is a random process; there will be a certain percentage of cells that are dividing in any time, which is different from the Fibonacci sequence [25]. Recently, the research of Viswanath et al indicated that RC for a random system will be approximately 1.133 [26,27]. It follows the patterns of random cell division, which has been confirmed by a mathematical modeling approach [20].…”

“…According to Fibonacci sequence analysis [25][26][27], we propose a method for AST that used the RC (recurrent coefficient for a growing sequence) of bacterial turbidity as the objective function, and therefore, the impact of composition heterogeneity of bacteria on AST will be well characterized. This method is expected to be developed into one commonly used method for pharmacodynamics in AST [28].…”