Slow Fibonacci Walks

TL;DR

This paper investigates the maximum steps in Fibonacci walks reaching a number n, disproving a conjecture about their distribution and providing explicit formulas and density results for various cases.

Contribution

It proves Stanley's conjecture holds for only a fraction of n, offers explicit formulas for initial pairs, and characterizes the number of steps in Fibonacci walks.

Findings

Stanley's conjecture is true for only a positive fraction of n.

Explicit formulas for initial pairs and step counts are provided.

Density results for distribution of cases are derived.

Abstract

For a positive integer $n$, we study the number of steps to reach $n$ by a {\it Fibonacci walk} for some starting pair $a_1$ and $a_2$ satisfying the recurrence of $a_{k+2}=a_{k+1}+a_k$. The problem of slow Fibonacci walks, first suggested by Richard Stanley, is to determine the maximum number $s(n)$ of steps for such a Fibonacci walk ending at $n$. Stanley conjectured that for most $n$, there is a slow Fibonacci walk reaching $n = a_s$ with the property that $a_{s+1}$ is the integer closest to $\phi n$ where $\phi=(1+\sqrt{5})/2$. We prove that this is true for only a positive fraction of $n$. We give explicit formulas for the choice of the starting pairs and the determination of $s(n)$ by giving a characterization theorem. We also derive a number of density results concerning the distribution of down and up cases (that is, those $n$ with $a_{s+1}=\lfloor \phi n\rfloor$ or $\lceil \phi n \rceil$, respectively), as well as for more general `paradoxical' cases.