Exercise 11.

You have access to \(1 \times 1\) tiles which come in 2 different colors and \(1\times 2\) tiles which come in 3 different colors. We want to figure out how many different \(1 \times n\) path designs we can make out of these tiles.

1. Find a recursive definition for the sequence \(a_n\) of paths of length \(n\text{.}\)

2. Solve the recurrence relation using the Characteristic Root technique.