Suppose you have $4$ types of tiles: blue dominoes, blue monominoes, red dominoes and red monominoes. Find the generating function for the number of ways of lining up tiles to cover length n such that:
a) no further restriction
b) all blue tiles, if any are before all red tiles.
My work. For part a), when I worked it out by actually sketching the possible combinations, I cancelled out the ones that would essentially be the same. For example, if $n=3$ there are three ways in which the board will be all blue which is $3x$ monominoes, and 2 tilings where theres $1x$ monomino and $1x$ domino, however since they all make the board all blue, wouldn't it simplify to being 1 combination? If I do this indefinitely, wouldn't this give me the generating function for the Fibonacci sequence?
For part b), I'm not too sure how to start this.