Problem solving episode---WalkTilings(1)

Here is the problem I am going to solve. It's called WalkTilings Problem.

Suppose you have tiles that are 1x2 rectangles, and you have to tile a walk that is 2x3 units long. You've got three choices: 3 vertical tiles, 1 vertical tile followed by two horizontal tiles, or two horizontal tiles followed by one vertical tiles.
How many choices do you have in tiling a 2x4 walk? How about a 2xn walk, where n is some natural number? Do you have a procedure that works for any n?

Here is the plan I want to work on.
Since it is hard to find out the solution when n is an abstract natural number at first, I will begin with the simplest condition(n = 1) and check several cases when n is small. I will investigate how the condition of n is related to the condition of n-1.

First, when n = 1, there is only one way to tile the walk.
When n = 2, there are two choices to tile the walk: 2 vertical tiles or 2 horizonal tiles.
When n = 3, there are three choices and they are presented in the problem.
When n = 4, there are 5 choices: 2 vertical tiles followed by 2 horizontal tiles, 4 vertical tiles, 1 vertical tile followed by 2 horizontal tiles and then followed by 1 vertical tile, 2 horizontal tiles followed by 2 vertical tiles, or 4 horizontal tiles.
When n = 5, there are 8 choices.