Problem 43179. Red and green towers

Frankie has a large number of wooden cubes, painted in red and green. He is building little towers with his cubes, by stacking them on top of each other.

He got bored after a while, so he introduced a rule: two red cubes may not be touching (see image).

Given n (the maximum number of cubes in a tower), determine the number m of different towers that Frankie can build.

Example: for n = 2, he can make 5 towers, so m=5:

1. [Red];
2. [Green];
3. [Red, Green];
4. [Green, Red];
5. [Green, Green].

You may assume that the input is an integer greater than zero.