The numbers of repeated palindromes in the Fibonacci and Tribonacci sequences

Abstract

The Fibonacci sequence $\mathbb{F}$ is the fixed point beginning with $a$ of morphism $\sigma(a,b)=(ab,a)$. Since $\mathbb{F}$ is uniformly recurrent, each factor $\omega$ appears infinite many times in the sequence which is arranged as $\omega_p$ $(p\ge 1)$. Here we distinguish $\omega_p\neq\omega_q$ if $p\neq q$. In this paper, we give algorithm for counting the number of repeated palindromes in $\mathbb{F}[1,n]$ (the prefix of $\mathbb{F}$ of length $n$). That is the number of the pairs $(\omega, p)$, where $\omega$ is a palindrome and $\omega_p\prec\mathbb{F}[1,n]$. 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 $\tau(a,b,c)=(ab,ac,a)$.