Find a generating function involving a tiling problem

Let ${h}_{n}$ be the number of ways to tile a $1\times n$ rectangle with $1\times 1$ tiles that are red or blue and $1\times 2$ tiles that are green, yellow, or white. Find a closed formula for $H(x)=\sum _{n\ge 0}{h}_{n}{x}^{n}.$

I am unsure how to start this problem. Is there a way to solve this without using recurrence relations?

Let ${h}_{n}$ be the number of ways to tile a $1\times n$ rectangle with $1\times 1$ tiles that are red or blue and $1\times 2$ tiles that are green, yellow, or white. Find a closed formula for $H(x)=\sum _{n\ge 0}{h}_{n}{x}^{n}.$

I am unsure how to start this problem. Is there a way to solve this without using recurrence relations?