In what I am currently doing, there naturally appears the following question: let $A$ be a square matrix with non-negative integer entries. Let $a_n$ be the sum of all entries of $A^n$.

Question: How the sequence $\{a_n\}_{n\geq 1}$ can grow?

Of course if $A$ is positive, then Perron-Frobenius Theorem tells us the answer, but in the general case of non-negative matrices, it can be difficult to guess the asymptotics of the sequence $\{A^n\}_{n\geq 1}$. So, I thought may be there is something known for this case, when we have actually integer matrices. Any references and comments would be appreciated.