In order to construct a bit string of length $n$, which contain $01$
$(1)$ we could start with $1$ and follow with a string length $n-1$ containing $01$
$(2)$ we could start with $0$ and follow with a string length $n-1$ containing $01$
$(3)$ we could start with $01$ and follow by any string length $n-2$
Therefore recurrence relation will be
$a_{n}=2a_{n-1}+2^{n-2}$