Paving a 2xn rectangle with 2 types of blocks

Question

pave a 2×n rectangle with blocks of two types, A and B, as illustrated (https://gyazo.com/0dac230cb94e167508b296f6e704b063)

Long edges are length 2 and short edges are length 1. I want to know in how many ways this can be done. My goal is to find a linear recursive equation that determines all possible combinations for a rectangle of length n

Experimentally I determined all possible combinations for the first 5 rectangles.

1. n = 0: x(n)=1
2. n = 1: x(n)=1
3. n = 2: x(n)=2
4. n = 3: x(n)=5
5. n = 4: x(n)=11

Answer 1

Let $R_n$ be the number of ways to pave the $2\times n$ rectangle and let $X_n$ be the number of way to pave a $2\times n$ rectangle that is missing one of the corners.

We are going to calculate each of these for $n\leq 2$ and then get the rest with a recursion.

for $n>2$ we have:

$R_n=R_{n-1}+R_{n-2}+2X_{n-1}$

$X_n=R_{n-1}+X_{n-1}$

These recursions are obtained by looking at the options for the right-most top block, and looking at the figure that is left unpaved.