# Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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## Random Fibonacci sequences and the number $1.13198824\dots$HTML articles powered by AMS MathViewer

by Divakar Viswanath
Math. Comp. 69 (2000), 1131-1155 Request permission

## Abstract:

For the familiar Fibonacci sequence (defined by $f_1 = f_2 = 1$, and $f_n = f_{n-1} + f_{n-2}$ for $n>2$), $f_n$ 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 random Fibonacci sequences defined by $t_1=t_2=1$, and for $n>2$, $t_n = \pm t_{n-1} \pm t_{n-2}$, where each $\pm$ sign is independent and either $+$ or $-$ with probability $1/2$ — it is not even obvious if $\vert {t_n}\vert$ should increase with $n$. Our main result is that \begin{equation*} \sqrt [n]{\vert {t_n}\vert } \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 measure, a computer calculation, and a rounding error analysis to validate the computer calculation.
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