The number of distinct and repeated squares and cubes in the Fibonacci sequence

2016/03/14 by Yuke Huang, Zhiying Wen, Huang, Yuke +2
Computer Science · Mathematics · #11B85 #68Q45 #Algorithms and Data Compression #Coding theory and cryptography #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:11B85 #msc:68Q45 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1603.04211

14 pages, 4 figures

arxiv created 2016/03/14 · openalex publication_date 2016/03/14 · arxiv updated 2016/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Fibonacci sequence \mathbbF is the fixed point beginning with a of morphism σ(a,b)=(ab,a). In this paper, we get the explicit expressions of all squares and cubes, then we determine the number of distinct squares and cubes in \mathbbF[1,n] for all n, where \mathbbF[1,n] is the prefix of \mathbbF of length n. By establishing and discussing the recursive structure of squares and cubes, we give algorithms for counting the number of repeated squares and cubes in \mathbbF[1,n] for all n, and get explicit expressions for some special n such as n=f_m (the Fibonacci number) etc., which including some known results such as in A.S.Fraenkel and J.Simpson, J.Shallit et al.

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