Number of binary strings of length $n$ satisfying specific (ad-hoc) conditions 
Count the number of binary strings of length $n$ that satisfy the following additional conditions:
a) Two zeroes in a row are not allowed
b) Three ones in a row are not allowed
c) The string cannot begin or end with $11$ (and of course because of a) above neither can it begin or end with $00$)


As a regular expression, the string matches ^1?(011?)*01?$ for $n>1$
The minimum number of zeroes is $\lfloor \frac{n+2}{3}\rfloor$. The maximum number of zeroes is $\lfloor \frac{n+1}{2}\rfloor$. For each count of zeroes, you can set up a base string consisting of alternating zeroes and ones, $0101\ldots10$, with $k$ zeroes, length $2k-1$, then insert $n-2k+1$ ones into the string in a choice of $k+1$ locations, ${k+1 \choose n-2k+1}$. Thus the answer is:
$$\sum_{k=\lfloor \frac{n+2}{3}\rfloor}^{\lfloor \frac{n+1}{2}\rfloor}{k+1 \choose n-2k+1}$$
