The Numbers of Distinct Squares and Cubes in the Tribonacci Sequence

TL;DR

This paper provides explicit formulas for counting the number of distinct squares and cubes within prefixes of the Tribonacci sequence, enhancing understanding of its combinatorial structure.

Contribution

It introduces explicit expressions for the counts of distinct squares and cubes in Tribonacci sequence prefixes, a novel combinatorial analysis.

Findings

Explicit formulas for the number of distinct squares in Tribonacci prefixes

Explicit formulas for the number of distinct cubes in Tribonacci prefixes

Enhanced understanding of the combinatorial structure of the Tribonacci sequence

Abstract

The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a,b,c)=(ab,ac,a)$. In this note, we give the explicit expressions of the numbers of distinct squares and cubes in $\mathbb{T}[1,n]$ (the prefix of $\mathbb{T}$ of length $n$).