Ways to fill a $n\times n$ square with $1\times 1$ squares and $1\times 2$ rectangles I came up with this question when I'm actually starring at the wall of my dorm hall.  I'm not sure if I'm asking it correctly, but that's what I roughly have:
So, how many ways (pattern) that there are to fill a $n\times n:n\in\mathbb{Z}_{n>0}$ square with only $1\times 1$ squares and $1\times 2$ rectangles?
For example, for a $2\times 2$ square:


*

*Four $1\times 1$ squares; 1 way.

*Two $1\times 1$ squares and one $1\times 2$ rectangle; $4$ ways total since we can rotate it to get different pattern.

*Two $1\times 2$ rectangles; 2 ways total: placed horizontally or vertically.
$\therefore$ There's a total of $1+4+2=\boxed{7}$ ways to fill a $2\times 2$ square.
So, I'm just wondering if there's a general formula for calculating the ways to fill a $n\times n$ square.
Thanks!

Two comments:
1 If you only allow for the 1x2 rectangles, the problem is known as the domino tiling problem. You can find the answer here: http://en.wikipedia.org/wiki/Domino_tiling#Counting_tilings_of_regions
2 In the case of a cheseboard (and I think it can be generalized to squares of even side length), there are few things known about the case with 2 1x1 squares and dominoes.
If the two squares are placed on two board squares of same color, it is impossible to complete the tiling, while if the 2 squares are placed on opposite colors Gomory's theorem says that it is always possible to complete the tiling (but I think it doesn't say in how many ways).
