Let $\lambda_1,\dots,\lambda_d$ be complex numbers that constitute the spectrum of a nonnegative integer matrix, and $P_1,\dots, P_d$ be complex polynoms, such that the sequence $$u_n=\Sigma_{i=1}^d P_i(n)\lambda_i^n$$ takes only real nonnegative values, i.e. $u_n\geq0, \forall n\in\mathbb N$.

I'm interested in the possible behaviours of $u_n$, and more precisely I would like to show the existence of an integer $k$ such that each sequence $(u_{nk+r})_{n\in\mathbb N}$ for $0\leq r\leq k-1$ is asymptotically equivalent to some $a_r n^{b_r}c_r^n$, where $a_r,b_r,c_r$ are real nonnegative constants depending on $r$.

For instance, $b_r=0$ and/or $c_r=1$are interesting cases that can happen.

Any hint would be appreciated, thanks !

**What I tried**

So I know that for reducible matrices, Perron-Froebenius theorem tells us that there is a real positive eigenvalue $\rho$ of maximal module, and that the other eigenvalues of maximal module are $\rho e^{\frac{ir\pi}{k}}$ with $0<r<k$. It shoud prevent counter-examples like the one from Gerald Edgar, but I'm missing the last argument to conclude. There are two problems in my opinions:

1) here the matrix is not necessarily irreducible,

2) the other eigenvalues (the ones that are not of maximal module) can also play a part.

Here is the most detailed paper I could find giving information about such eigenvalues, but I don't see how to use this information: The spectra of nonnegative integer matrix via formal power series.