
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 , we study the number of steps to reach by a {\it Fibonacci walk} for some starting pair and satisfying the recurrence of . The problem of slow Fibonacci walks, first suggested by Richard Stanley, is to determine the maximum number of steps for such a Fibonacci walk ending at . Stanley conjectured that for most , there is a slow Fibonacci walk reaching with the property that is the integer closest to where . We prove that this is true for only a positive fraction of . We give explicit formulas for the choice of the starting pairs and the determination of by giving a characterization theorem. We also derive a number of density results concerning the distribution of down and up cases (that is, those with or $\lceil \phi…
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