powers of $\frac{1+\sqrt a}2$ 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 $c_n=\frac{a-1}4b_{n-1}$

If $a \equiv 5$ ($\bmod 8$) then this happens infinitely many times.  This follows from the relation $x^2 = x + \tfrac{a-1}{4}$ and $\tfrac{a-1}{4}$ is odd.  Suppose $x^n = b_nx + c_n$ for integers $b_n,c_n$ then
$$
x^{n+1} = (b_n+c_n)x + b_n\frac{a-1}{4}
$$
So $b_n \equiv 1, 1, 0, 1, 1, 0, \dotsc$ ($\bmod 2$).
