This is a more general version of a question I have asked before. Let $a_1,\dots,a_r\in\mathbb C$ be algebraic numbers and suppose that for every natural number $n$ the sum $$ F(n)=\sum_{j=1}^r a_j^n $$ is a nonnegative integer. What can be said about the asymptotics of the sequence $F(n)$? For instance, if $a_j=Re^{2\pi ij/r}$, then $F(n)$ is zero except for the case when $n=rm$ for a natural $m$ and then $F(rm)=rR^{rm}$. But what can be said generally? Do the conditions force in that the $a_j$ may be grouped together in a way that each part satisfies the conditions of the example? Or at least the leading one does?

In general, the $a_i$s can't be grouped as in your example, but something not too dissimilar happens. I'll start with two examples when $r=2$. Example 1. Take and $a_1,a_2$ as the roots of the equation $x^2+2x+10$: they are complex conjugate with $a_1 = a_2 = \sqrt{10}$. Example 2. Take $a_1 > a_2$ to be the roots of $x^25x+2$, then $F(n)$ grows like $a_1^n$. I would expect the generic behaviour to be similar to one of the two examples above. In general, your $a_i$s will be the set of solutions to a (monic) polynomial $P\in\mathbb{Z}[x]$; generically, the largest roots (in absolute value) of $P$ will be either a pair of conjugate complex roots or a single real root. In the second case, the asymptotics is clear, like in Example 2 above; in the first case, one gets pretty much the same as Example 1. To be more precise than I have been above, one should understand how small $\cos(n\theta\pi)$ can be: I would expect that generically this is never smaller than $1/n^k$ for some $k$, which would imply that the sequence $F(n)$ is asymptotically "between" $2\rho^n/n^k$ and $2\rho^n$ (up to lower order terms). 

