Fibonacci Sequences of 1D, 2D Words: Enumerating and Locating the Factors of the Fixed Points

TL;DR

This paper explores methods to enumerate and locate factors within the infinite Fibonacci array, analyzing their complexity and distribution to better understand the structure of these two-dimensional words.

Contribution

It introduces two enumeration methods and a locating technique for factors in Fibonacci arrays, along with an analysis of their complexity and distribution.

Findings

Two enumeration methods for Fibonacci array factors

A technique for locating factors within the array

Analysis of factor complexity and distribution

Abstract

Given an infinite word, enumerating its factors is an important exercise for understanding the structure of the word. The process of finding all the factors is quite tricky for two-dimensional words. In this paper, two possible ways of enumerating the factors of the fixed point ($f_{\infty,\infty}$) of the sequence of Fibonacci arrays and a method for locating these factors in $f_{\infty,\infty}$ are explored. In addition, the factor complexity and the locations of the factors of the fixed point of Fibonacci sequence of arrays are also analysed.