Define a sequence by $a_1=1$ and$$a_{n+1}=\lfloor \sqrt{2}(a_n+1/2)\rfloor$$for $n\ge 1$. The difference $a_{2n+1}-2a_{2n-1}$ is the $n$th digit in the binary expansion of $\sqrt{2}$.Find similar results for $\theta =\sqrt{m}$, and other algebraic numbers.