Problem #WSP-000268

Problem

How many subsets are there of $\{1,2,...,10\}$ (the integers from $1$ to $10$ inclusive) containing no consecutive digits? That is, we do count $\{1,3,6,8\}$ but do not count $\{1,3,6,7\}$.
For example, when $n=3$, we have $8$ subsets overall but only $5$ contain no consecutive integers. The $8$ subsets are $\varnothing$ (the empty set), $\{1\}$, $\{2\}$, $\{3\}$, $\{1,3\}$, $\{1,2\}$, $\{2,3\}$ and $\{1,2,3\}$, but we exclude the final three of these.