The numbers of powers in the Tribonacci sequence

The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $σ(a)=ab$, $σ(b)=ac$, $σ(c)=a$. 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 $α$-powers in $\mathbb{T}[1,n]$ for $α\geq2$ and $n\geq1$.

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