The Square Trees in the Tribonacci Sequence

TL;DR

This paper explores the structure and enumeration of repeated squares in the Tribonacci sequence, providing explicit formulas, a tree-based framework, and efficient algorithms for counting repeated squares in its prefixes.

Contribution

It introduces the concept of square trees to analyze the positions of repeated squares and develops a fast counting algorithm for these squares in the Tribonacci sequence.

Findings

Explicit expressions for all squares in the Tribonacci sequence

A tree structure (square trees) for positions of repeated squares

A fast algorithm for counting repeated squares in prefixes

Abstract

The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a,b,c)=(ab,ac,a)$. In this note, we get the explicit expressions of all squares, and then establish the tree structure of the positions of repeated squares in $\mathbb{T}$, called square trees. Using the square trees, we give a fast algorithm for counting the number of repeated squares in $\mathbb{T}[1,n]$ for all $n$, where $\mathbb{T}[1,n]$ is the prefix of $\mathbb{T}$ of length $n$. Moreover we get explicit expressions for some special $n$ such as $n=t_m$ (the Tribonacci number) etc.