  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).
