The numbers of powers in the Tribonacci sequence

2016/09/21 by Yuke Huang, Huang, Yu-Ke, Zhi‐Ying Wen +1
Physics and Astronomy · Computer Science · Mathematics · #Advanced Mathematical Theories and Applications #semigroups and automata theory #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.1609.06471

Abstract

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

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