Creating a $2\times n$ rectangle out of two block type The question:
I want to pave a $2\times n$ rectangle with blocks of two types, A and B, as illustrated 
long edges are length $2$ and short edges are length $1$. I want to know in how many ways this can be done. Reflections of combinations DO count as separate combinations.
(a) Find a linear recursive equation for $X_n$, the number of pavings of a $2\times n$ rectangle.
I have no idea what to do except start by placing blocks on the left-side of the $2\times n$ rectangle, but I don't know what else I can do.

You have $X_n$ as the number of ways to pave a $2 \times n$ rectangle.  At the right hand end you might have a type B piece, a vertical type A piece, or two horizontal type A pieces.  If you take off the A piece(s) you are left with a tiled rectangle.  If you take off a B piece you are left with a tiled rectangle with one extra square, so define $Y_n$ at the number of ways to tile a $2 \times n$ rectangle plus the top square of the next column, which is the same as the number of ways to tile the rectangle plus the bottom square of the next column.  One of these rectangles plus square can either have a B piece attached to a rectangle, or a horizontal A piece sticking out.  This should suggest a set of coupled recurrences for $X_n,Y_n$.
