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arXiv · 1609.06471

The numbers of powers in the Tribonacci sequence

Yu-Ke Huang·Zhi-Ying Wen
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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 $n\geq1$.

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Yu-Ke Huang, Zhi-Ying Wen. 2016-09-21. The numbers of powers in the Tribonacci sequence. https://arxiv.org/abs/1609.06471

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