Let \(a_n\) be the number of \(1 \times n\) tile designs you can make using \(1 \times 1\) squares available in 4 colors and \(1 \times 2\) dominoes available in 5 colors.
First, find a recurrence relation to describe the problem. Explain why the recurrence relation is correct (in the context of the problem).
Write out the first 6 terms of the sequence \(a_1, a_2, \ldots\text{.}\)
Solve the recurrence relation. That is, find a closed formula for \(a_n\text{.}\)