Slow Recurrences

TL;DR

This paper generalizes the concept of slow Fibonacci-like sequences to arbitrary positive parameters, providing a characterization theorem and analyzing the total number and slowest variants of such walks.

Contribution

It introduces a comprehensive framework for $(eta,eta)$-walks, extending previous studies on Fibonacci sequences, and derives new results on their enumeration and extremal properties.

Findings

Characterization theorem for $(eta,eta)$-walks.

Formulas for counting $n$-slow walks for given $n$.

Identification of the slowest $n$-slow walk among all parameters.

Abstract

For positive integers $\alpha$ and $\beta$, we define an $(\alpha,\beta)$-walk to be any sequence of positive integers satisfying $w_{k+2}=\alpha w_{k+1}+\beta w_k$. We say that an $(\alpha,\beta)$-walk is $n$-slow if $w_s=n$ with $s$ as large as possible. Slow $(1,1)$-walks have been investigated by several authors. In this paper we consider $(\alpha,\beta)$-walks for arbitrary positive $\alpha,\beta$. We derive a characterization theorem for these walks, and with this we prove several results concerning the total number of $n$-slow walks for a given $n$. In addition to this, we study the slowest $n$-slow walk for a given $n$ amongst all possible $\alpha,\beta$.