For any a which is not a perfect square, let $x=\frac{1+\sqrt a}2$.
$x^n$ can be written uniquely as $b_nx+c_n$, where b and c are rational.
Apart from $a=0, a=1, a= 1 \pm 2^m$ for $m>2$, are there any other values of $a$ for which $b$ or $c$ is an integer for infinitely many $n$? If not, are there any upper bounds on the values of n for which $b$ or $c$ is an integer?
e.g for $a=7$
$\\b \ c\\ 0 \ 1\\ 1 \ 0\\ 1 \ \frac{3}2\\ \frac{5}2 \ \frac{3}2\\ 4 \ \frac{15}2\\ \frac{23}2 \ 6\\ \frac{35}2 \ \frac{69}4$
$b_n=b_{n-1}+c_{n-1}$ and $c_n=\frac{a-1}4b_{n-1}$