AIME 2015 Problem 12

We can define the number of n-letter strings that contain no more than three $a's$ or $b's$ in a row as $s_{n}$ . We can piece together that $s_{1}=2, s_{2}=4, s_{3}=8$ , and since there are $2$ 3-letter strings that contain 3 $a's$ and 3 $b's$ , then $s_{4}=16-2=14$ .

Looking at our sequences, to find the value of $s_{n}$ , we can tack on either an $a$ or a $b$ at the end of our previous sequence. This however wont always work. If $s_{n-1}$ ends in 3 $a's$ or $b's$ in a row, then only one of the two letters will work. Given any sequence of length $n-4$ , we can always add three of whatever letter that sequence doesn't end in, so there are $s_{n-4}$ sequences such that we can only add on one of the two letters at the end.

This gives us the formula $s_{n}=2s_{n-1}-s_{n-4}$ and using the above information, we can eventually arrive at the answer of $\boxed{548}$