jmc

Let $a_n$ and $b_n$ denote, respectively, the number of sequences of length $n$ ending in $A$ and $B$. If a sequence ends in an $A$, then it must have been formed by appending two $A$s to the end of a string of length $n-2$. If a sequence ends in a $B,$ it must have either been formed by appending one $B$ to a string of length $n-1$ ending in an $A$, or by appending two $B$s to a string of length $n-2$ ending in a $B$. Thus, we have the recursions$$\begin{array}{rl}{a}_n& =a_{n-2}+b_{n-2}\\ b_n& =a_{n-1}+b_{n-2}\end{array}$$By counting, we find that $a_1=0,b_1=1,a_2=1,b_2=0$.$$\begin{array}{rrrrrr}n& a_n& b_n& n& a_n& b_n\\ 1& 0& 1& 8& 6& 10\\ 2& 1& 0& 9& 11& 11\\ 3& 1& 2& 10& 16& 21\\ 4& 1& 1& 11& 22& 27\\ 5& 3& 3& 12& 37& 43\\ 6& 2& 4& 13& 49& 64\\ 7& 6& 5& 14& 80& 92\end{array}$$Therefore, the number of such strings of length $14$ is $a_{14}+b_{14}=\boxed{172}$.