TTsphn wrote:

$c = xy,b = x + y$
We has $a_{n + 2} = ba_{n + 1} - ca_n$
We have :
$a_{n + 1}^2 - a_na_{n + 2} = c^n$
We prove by induction.
Call $a_m,a_{m + 1},a_{m + 2},a_{m + 3}$ is four consective integer in this sequence.
$c^m = a_{m + 1}^2 - a_ma_{m + 2}$
$c^{m + 1} = a_{m + 2}^2 - a_{m + 1}a_{m + 3}$
Imply that $c\in Q$
Because $c^m\in Z$ so $c\in Z$
Suppose $c=\frac{p}{q}$ where $\gcd(p,q)=1$ so $q^m|p^m\Rightarrow q=1$ so $c\in Z$
$b = \frac {a_{m + 2} - ca_m}{a_{m + 1}}\Rightarrow b\in Q$
The general of sequence imply that b is a rational root of a monic Polynomial .(Because $a_n = f_n(b)$ )
so $b\in Z$
imply that $b,c\in Z$
We has $a_1 = 1,a_2 = b$ so $a_n\in Z,\forall n\in N$

What is the monic polynomial?