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##### Department of Mathematics, University of California San Diego

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## Slow Fibonacci Walks

##### Abstract:

We define a Fibonacci walk to be any sequence of positive integers satisfying the recurrence $w_{k+2}=w_{k+2}=w_{k+1}+w_k$, and we say that a sequence is an $n$-Fibonacci walk if $w_k=n$ for some $k$. Note that every $n$ has a number of (boring) $n$-Fibonacci walks, e.g. the sequence starting $n,n,2n,\ldots$. To make things interesting, we consider $n$-Fibonacci walks which have $w_k=n$ with $k$ as large as possible, and we call this an $n$-slow Fibonacci walk. For example, the two 6-slow Fibonacci walks start 2, 2, 4, 6 and 4, 1, 5, 6. In this talk we discuss a number of properties about $n$-slow Fibonacci walks, such as the number of slow walks a given $n$ can have, as well as how many $n$ have a given number of walks. We also discuss slow walks that follow more general recurrence relations. This is joint work with Fan Chung and Ron Graham.

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