The structure of palindromes in the Fibonacci sequence and some applications

TL;DR

This paper explores the structural properties of palindromes within the Fibonacci sequence, establishing three key structures, and uses them to analyze palindrome occurrences, providing algorithms and proofs for related combinatorial properties.

Contribution

It introduces three fundamental structures of Fibonacci palindromes and applies them to count and analyze palindrome occurrences, including new algorithms and simplified proofs.

Findings

Number of distinct palindrome occurrences in the first n Fibonacci sequence elements is n

Provided an algorithm to count repeated palindrome occurrences

Derived explicit formulas for specific sequence lengths

Abstract

Let ${\cal P}$ be the set of palindromes occurring in the Fibonacci sequence. In this note, we establish three structures of $\mathcal{P}$ and and discuss their properties: cylinder structure, chain structure and recursive structure. Using these structures, we determine that the number of distinct palindrome occurrences in $\mathbb{F}[1,n]$ is exactly $n$, where $\mathbb{F}[1,n]$ is the prefix of the Fibonacci sequence of length $n$. Then we give an algorithm for counting the number of repeated palindrome occurrences in $\mathbb{F}[1,n]$, and get explicit expressions for some special $n$, which include the known results. We also give simpler proofs of some classical properties, such as in X.Droubay, W.F.Chuan and J.Shallit et al.