The numbers of distinct and repeated squares and cubes in the Tribonacci sequence

TL;DR

This paper analyzes the Tribonacci sequence to provide explicit formulas and algorithms for counting the number of distinct and repeated squares and cubes within its prefixes, advancing combinatorial understanding of this sequence.

Contribution

It offers explicit formulas and algorithms for counting and analyzing squares and cubes in the Tribonacci sequence, including special cases like Tribonacci numbers.

Findings

Explicit expressions for the number of distinct squares and cubes.

Algorithms for counting repeated squares and cubes.

Results for special lengths such as Tribonacci numbers.

Abstract

The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a,b,c)=(ab,ac,a)$. The main result is twofold: (1) we give the explicit expressions of the numbers of distinct squares and cubes in $\mathbb{T}[1,n]$ (the prefix of $\mathbb{T}$ of length $n$); (2) we give algorithms for counting the number of repeated squares and cubes in $\mathbb{T}[1,n]$ for all $n$; then get explicit expressions for some special $n$ such as $n=t_m$ (the Tribonacci number).