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Mathematics of Computation

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Random Fibonacci sequences
and the number $1.13198824\ldots$

Author: Divakar Viswanath
Journal: Math. Comp. 69 (2000), 1131-1155
MSC (1991): Primary 15A52, 28A80, 60J05, 65G05
Published electronically: June 10, 1999
MathSciNet review: 1654010
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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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Additional Information

Divakar Viswanath
Affiliation: Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, California 94720

Received by editor(s): November 17, 1997
Received by editor(s) in revised form: July 14, 1998
Published electronically: June 10, 1999
Additional Notes: This work was supported in part by NSF Grant DMS-9500975CS and DOE Grant DE-FG02-94ER25199 to Lloyd N. Trefethen.
Article copyright: © Copyright 2000 American Mathematical Society