Consider a $2 \times n$ grid that is to be tiled with up to $n$ tiles of dimensions $2\times 1$ and up to two tiles of dimensions $2\times 2$. Rotation of tiles is not allowed. In how many ways can this tiling of the grid be done?
My work
I was able to work out a recurrence to find $T_n$, the number of ways to tile a $2\times n$ grid with tiles of given dimensions without constraints on number of tiles used.
$$T_n = T_{n-1} + 2T_{n-2}$$ with $T_1=1$, $T_2=2$
However I am unable to work out the constraint on the number of tiles.
Any help is appreciated.
Because you cannot rotate, the $2\times$ might as well be $1\times$ throughout. Once you place the big tiles, there is only one way to complete the tiling, so you can just count the numbers of packings of $0$, $1$, or $2$ big tiles. For $0$ big tiles, there is only $1$. For $1$ big tile, you have $n-1$ choices. For $2$ big tiles, you have $\binom{n-1}{2}-(n-2)$ choices if $n \ge 2$.