TITLE: Recursively Defined Entities
QUESTION [5 upvotes]: So I am having some trouble understanding how one is to come up with the recursive  definition to the following problem...
We are given a rectangle of width $2$ and length $n$. Suppose we have dominoes of size $2\times 1$. What is the number of different ways we can cover the $2\times n$ rectangle? 
The solution is suppose to be $a(n) = a(n-1) + a(n-2)$ but I'm not understanding exactly how they have arrived at that. I can plug values in and see that it does indeed work but what the intuition is behind that is what confuses me. 
Thanks in advance!

REPLY [5 votes]: In a corner, you have two possibilities to place a domino, either vertical
|x|...
|x|...

which leaves you with an $(n-1)×2$ rectangle to tile, or horizontally
xx...
.....

which forces a horizontal domino below
xx...
yy...

and leaves you with an $(n-2)×2$ rectangle.