Powers of a Toeplitz matrix I'm searching a closed formula to compute the powers of the following matrix
\begin{equation*}F\triangleq \begin{bmatrix}
1 & T & \frac{T^2}{2}\\
0 & 1 & T\\
0 & 0 & 1
\end{bmatrix}\end{equation*}
where $T$ is a given parameter. For example, if I'm not wrong the first 5+1 powers are
\begin{equation*}F^0\triangleq \begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1
\end{bmatrix}\end{equation*}
\begin{equation*}F^1\triangleq \begin{bmatrix}
1 & T & \frac{T^2}{2}\\
0 & 1 & T\\
0 & 0 & 1
\end{bmatrix}\end{equation*}
\begin{equation*}F^2\triangleq \begin{bmatrix}
1 & 2T & 2T^2\\
0 & 1 & 2T\\
0 & 0 & 1
\end{bmatrix}\end{equation*}
\begin{equation*}F^3\triangleq \begin{bmatrix}
1 & 3T & \frac{9T^2}{2}\\
0 & 1 & 3T\\
0 & 0 & 1
\end{bmatrix}\end{equation*}
\begin{equation*}F^4\triangleq \begin{bmatrix}
1 & 4T & 8T^2\\
0 & 1 & 4T\\
0 & 0 & 1
\end{bmatrix}\end{equation*}
\begin{equation*}F^5\triangleq \begin{bmatrix}
1 & 5T & \frac{25T^2}{2}\\
0 & 1 & 5T\\
0 & 0 & 1
\end{bmatrix}\end{equation*}
From these results it seems to me that for any $k$ holds
\begin{equation*}F^k=\begin{bmatrix}
1 & kT & \frac{(kT)^2}{2} \\
0 & 1 & kT \\
0 & 0 & 1
\end{bmatrix}\end{equation*}
is it true? If yes, how can be proved my formula?

Your matrix can be written as an exponential of a nilpotent matrix:
$$F=e^{TN},\qquad N=\left(\begin{array}{ccc} 0 & 1 & 0 \\
0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right).$$
Note that $N^2=\left(\begin{array}{ccc} 0 & 0 & 1 \\
0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right)$ and $N^3=0$. Now,
$$F^k=e^{kT N}=\mathbf 1+kTN+\frac{k^2T^2}{2}N^2,$$
which gives the result.
P.S. If you are uncomfortable with matrix exponentials, then simply use induction.
