The numbers of powers in the Tribonacci sequence

TL;DR

This paper investigates the occurrence and counting of powers like squares and cubes within the Tribonacci sequence, providing explicit formulas, algorithms, and discussions on higher powers for sequence prefixes.

Contribution

It offers explicit formulas and algorithms for counting squares and cubes in Tribonacci sequence prefixes, and explores higher powers, advancing combinatorial understanding.

Findings

Explicit formulas for counts of squares and cubes

Algorithms for counting repeated powers

Discussion on higher powers in the sequence

Abstract

The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a)=ab$, $\sigma(b)=ac$, $\sigma(c)=a$. The prefix of $\mathbb{T}$ of length $n$ is denoted by $\mathbb{T}[1,n]$. The main result is threefold, we give: (1) explicit expressions of the numbers of distinct squares and cubes in $\mathbb{T}[1,n]$; (2) algorithms for counting the numbers of repeated squares and cubes in $\mathbb{T}[1,n]$; (3) a discussion about $\alpha$-powers in $\mathbb{T}[1,n]$ for $\alpha\geq2$ and $n\geq1$.