Building on Ross's hint.
Let $a_n$ be the number of good strings of length $n$
Let $b_n$ be the number of good strings of length $n$ which do not end in a $1$ or $2$
Let $c_n$ be the number of good strings of length $n$ which end in $1$
Let $d_n$ be the number of good strings of length $n$ which end in $2$
This gives the recurrence $a_n=b_n+c_n+d_n$
Obviously, $b_n = 5a_{n-1}$
Also we have $c_n=b_{n-1}+d_{n-1}$ because the right hand side of the equation is the number of good strings of length $n-1$ which do not end in a $1$.
Finally, $d_n=b_{n-1}+c_{n-1}$ because the right hand side of the equation is the number of good strings of length $n-1$ which do not end in a $2$
Substituting for $b_n$, $c_n$, $d_n$ in the first equation the equations we get, $a_n = 5a_{n-1}+b_{n-1}+d_{n-1}+b_{n-1}+c_{n-1}$ = $5a_{n-1}+b_{n-1}+a_{n-1}$ = $6a_{n-1}+5a_{n-2}$
Therefore, $a_n=6a_{n-1}+5a_{n-2}$