Let $r≥4$ be a positive intger. Let us consider the difference non-autonomous equation: $$u_{n+1}=(1+r^{2n+1})u_{n}-r^{2n-1}u_{n-1}+2 \tag{*}$$
All solutions of $(*)$ have the form: $$u_{n}=\sum_{m=1}^{n}\left(2(m-1)+u_1-ru_0\right)r^{n^2-m^2}+u_0r^{n^2}$$ where $u_0,u_1$ are the initial conditions.
I have the following questions:
1) Show that all solutions of (*) tends toward infinity.
2) Find sufficient and necessary conditions in which $u_{n}$ is an integer
Factoring, $$u_n = r^{n^2} \left( u_0 + \sum_{m=1}^n (2m-2+u_1-r u_0) r^{-m^2} \right)$$
If $r > 1$ then by the ratio test the sum converges as $n \to \infty$, and so $u_n \to \infty$ because it is the limit of $r^{n^2}$ times a constant. To see this more easily, note that $\sum_{m=1}^{\infty} a r^{-m^2}$ converges by the ratio test, so we only actually have to consider $\sum_{m=1}^{\infty} m r^{-m^2}$. That converges also by the ratio test.
Now, $u_n$ is an integer if $u_0$ and $u_1$ are, by the recurrence equation. Indeed, if two successive $u_i$ are both integer then all remaining $u_i$ are.
I have yet to find a necessary condition.