I'm certain the pattern the system creates is
$$ A^kX(0) = \begin{pmatrix}2^k\\1\\2^k\end{pmatrix}\hspace{3pc} $$
Where A is a matrix created by the system and X(0) is a solution vector at t=0
Im going to omit other info since it is not needed.
The rest of the answer I'm not to sure about. Am I correct that the given pattern will then mean: $$ X(t) = \sum_{k=0}^{+\infty} \frac{2^k}{k!}t^k $$
That is
$$ X(t) = e^t\begin{pmatrix}2^k\\1\\2^k\end{pmatrix}\hspace{3pc} $$
?