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Random Fibonacci sequences and the number 1.13198824...

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\begin{abstract} For the familiar Fibonacci sequence --- defined by f1=f2=1, and fn=fn−1+fn−2 for ngreaterthan2 --- 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 {\it random} Fibonacci sequences defined by t1=t2=1, and for ngreaterthan2, tntn−1±tn−2, where each ± sign is independent and either + or with probability 1/2 --- it is not even obvious if \abstn 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… 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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1997-10

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Cornell University

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computer science; technical report

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

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technical report

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