What are the 2-adic numbers beginning $\overline{10}$ and no consecutive two numbers permissible?
i.e.:
$\overline{10}$
$\overline{10}1$
$\overline{10}11$ etc.
But if we allow $\overline{10}11$ then $\overline{10}10$ is not permitted.
I expect there to be two mutually exclusive sets although I may be mistaken.
As for where I've got to with this problem... I really have no idea how some rule governing the repeating part to the left affects the set of numbers possible.