The numbers of powers in the Tribonacci 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 $\alpha \ge 2$ and $n\ge 1$.

17 pages, 4 figures