The numbers of repeated palindromes in the Fibonacci and Tribonacci sequences

TL;DR

This paper develops algorithms to count repeated palindromes in Fibonacci and Tribonacci sequences, providing explicit formulas for specific lengths and advancing understanding of their combinatorial structures.

Contribution

It introduces algorithms for counting repeated palindromes in Fibonacci and Tribonacci sequences and derives explicit formulas for certain sequence lengths.

Findings

Algorithms for counting repeated palindromes in Fibonacci sequences

Explicit formulas for counts at Fibonacci sequence lengths

Extension of results to 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 $\tau(a,b,c)=(ab,ac,a)$.