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## Random Fibonacci sequences and the number 1.13198824...

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**Author**

Viswanath, Divakar

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**Abstract**

\begin{abstract} For the familiar Fibonacci sequence --- defined by $f_1 = f_2 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n greater than 2$ --- $f_n$ increases exponentially with $n$ at a rate given by the golden ratio $(1+\sqrt{5})/2=1.61803398\ldots$. But for a simple modification with both additions and subtractions --- the {\it random} Fibonacci sequences defined by $t_1=t_2=1$, and for $n greater than 2$, $t_n = \pm t_{n-1} \pm t_{n-2}$, where each $\pm$ sign is independent and either $+$ or $-$ with probability $1/2$ --- it is not even obvious if $\abs{t_n}$ should increase with $n$. Our main result is that \begin{equation*} \sqrt[n]{\abs{t_n}} \rightarrow 1.13198824\ldots\:\:\: \text{as}\:\:\: n \rightarrow\infty \end{equation*} with probability $1$. Finding the number $1.13198824\ldots$ involves the theory of random matrix products, Stern-Brocot division of the real line, a fractal-like measure, a computer calculation, and a rounding error analysis to validate the computer calculation. \end{abstract}

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**Date Issued**

1997-10#####
**Publisher**

Cornell University

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**Subject**

computer science; technical report

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**Previously Published As**

http://techreports.library.cornell.edu:8081/Dienst/UI/1.0/Display/cul.cs/TR97-1650

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**Type**

technical report