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 $00$)


mmm - I probably posted the question too quickly, anyway in last few hours, here is I think the solution in terms of recurrence relations:
Let $N(n)$ be the number of binary strings satisfying the conditions.
Let $N_1(n)$ be the number of binary strings ending in a $1$ satisfying the conditions.
Let $N_0(n)$ be the number of binary strings ending in a $0$ satisfying the conditions. Then 
$$N_0(n) = N_1(n-1) + N_0(n-3)$$
$$N_1(n) = N_0(n-1)$$
$$N(n) = N(n-2) + N(n-3)$$
with $N_0(1) = N_1(1) = N_0(2) = N_1(2) = N_0(3) = N_1(3)=1$.
