The numbers of repeated palindromes in the Fibonacci and Tribonacci sequences

2016/04/18 by Yuke, Huang, Zhiying, Wen
#11B85 #68Q45 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.05021

Abstract

The Fibonacci sequence \mathbbF is the fixed point beginning with a of morphism σ(a,b)=(ab,a). Since \mathbbF is uniformly recurrent, each factor ω appears infinite many times in the sequence which is arranged as ω_p (p≥ 1). Here we distinguish ω_p≠ω_q if p≠ q. In this paper, we give algorithm for counting the number of repeated palindromes in \mathbbF[1,n] (the prefix of \mathbbF of length n). That is the number of the pairs (ω, p), where ω is a palindrome and ω_p\prec\mathbbF[1,n]. We also get explicit expressions for some special n such as n=f_m (the m-th Fibonacci number). The similar results are also given to the Tribonacci sequence, the fixed point beginning with a of morphism τ(a,b,c)=(ab,ac,a).

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