Slow Recurrences

2019/09/14 by Sam Spiro, Spiro, Sam
Computer Science · Mathematics · #11B39 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1909.06517

openalex publication_date 2019/09/14 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

For positive integers α and β, we define an (α,β)-walk to be any sequence of positive integers satisfying w_k+2=αw_k+1+βw_k. We say that an (α,β)-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 (α,β)-walks for arbitrary positive α,β. 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 α,β.

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