If no $\times10$ steps, we get $1$ routes. If $1$ step of $\times 10$ then there are $11$ routes since this step happens before you exceed $11$. If $2$ step of $\times 10$ then there are $2$ routes since the first such step must happen immediately. There are no routes involving more than $2$ steps of $\times 10$. That is a total of $14$ routes. This is also how many times $111$ happens since every positive integer is uniquely reachable from $1$ by a sequence of $n\mapsto 2n$ and $n\mapsto 2n+1$ moves. The indices where $111$ happens are: $14$ $4098$ $14336$ $2099200$ $1075838976$ $551903297536$ $283673999966208$ $146366987889541120$ $76092819304051900416$ $40140115104391984316416$ $21760664753063325144711168$ $12379400392853802748991242240$ $7605903601369376408980219232256$ $1298074214633706907132624082305024$