Given a $2^k \times 2^k$ grid with $k \geq 1$, it's well known that it's possible to cover the grid minus one (arbitrary) point with $L$-trominoes. The gist of the proof involves considering the four $2^{k-1} \times 2^{k-1}$ quadrants of the grid and arguing by induction.
I'm wondering whether or not there are known results on the number of such tilings (given the point we want to exclude).