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](http://www.ams.org/journals/jams/2000-13-04/S0894-0347-00-00342-8/S0894-0347-00-00342-8.pdf).



A complex sequence with positive values
No.


Consider the matrix
$$\begin{pmatrix}
7 & 8 & 4 & 4 & 0 \\
0 & 7 & 4 & 4 & 0 \\
4 & 4 & 7 & 8 & 0 \\
4 & 4 & 0 & 7 & 0 \\
0 & 0 & 0 & 0 & 5 \\
\end{pmatrix}$$
The eigenvalues are $3 \pm 4i$, $11 \pm 4 \sqrt{3}$ and $5$. So
$$u\_n = (3+4i)^n + (3-4i)^n + 3 \cdot 5^n + 0 \cdot (11+4 \sqrt{3})^n + 0 \cdot (11-4 \sqrt{3})^n$$
is of the required form but does not have the desired asymptotics. (Example found using the method of Geoff Robinson's answer [here](https://mathoverflow.net/a/112535/297).) 


Requiring that all the $P\_i$ be nonzero doesn't help. Let $M$ be the above matrix and let $N$ be the $10 \times 10$ matrix with block form 
$$\begin{pmatrix} 0 & M^2 \\ \mathrm{Id} & 0 \end{pmatrix}.$$
The eigenvalues of $N$ are $\pm3 \pm 4 i$, $\pm 11 \pm 4 \sqrt{3}$ and $\pm 5$.


Now look at
$$10000 \cdot \left( (11+4 \sqrt{3})^n + (-11-4 \sqrt{3})^n +(-11+4 \sqrt{3})^n +(11-4 \sqrt{3})^n \right) $$
$$+100 \cdot 5^n + (-5)^n + 2 ((3+4i)^n + (3-4i)^n) + ((-3+4i)^n + (-3-4i)^n)$$


Of course, the exact powers of $10$ don't matter. The point is that $10000$ is large enough to always make things positive for $n$ even. For $n$ odd, the first term vanishes and $100$ is large enough to make the remaining terms positive, but $(3+4i)^n$ is of the same magnitude as $5^n$, so you can't get an asymtotic of the sort you want. 


