Solution:
Let $a_{n}$ be the number of binary sequences of length $n$ satisfying the conditions and ending in $0$, let $b_{n}$ be the number ending in $01$, and let $c_{n}$ be the number ending in $11$. From the legal sequences of length $2$ ($01$, $11$, $10$), we find that $a_{2}=b_{2}=c_{2}=1$. We now establish a recursion by building sequences of length $n+1$ from sequences of length $n$.

We can add a $0$ to a sequence of length $n$ if and only if it ended with a $1$, so $a_{n+1}=b_{n}+c_{n}$.

We can have a sequence of length $n+1$ ending with $01$ only by adding a $1$ to a sequence of length $n$ ending in $0$, so $b_{n+1}=a_{n}$.

We can have a sequence of length $n+1$ ending with $11$ only by adding a $1$ to a sequence of length $n$ ending in $01$, so $c_{n+1}=b_{n}$.

We can now run the recursion: