The Square Trees in the Tribonacci Sequence

Abstract: The Tribonacci sequence $\mathbb{T}$ is the fixed point of the substitution $\sigma(a,b,c)=(ab,ac,a)$. In this note, we get the explicit expressions of all squares, and then establish the tree structure of the positions of repeated squares in $\mathbb{T}$, called square trees. Using the square trees, we give a fast algorithm for counting the number of repeated squares in $\mathbb{T}[1,n]$ for all $n$, where $\mathbb{T}[1,n]$ is the prefix of $\mathbb{T}$ of length $n$. Moreover we get explicit expressions for some special $n$ such as $n=t_m$ (the Tribonacci number) etc.