If I have a nx2 grid which I need to fill using 2x1 dominoes and L shaped trominoes in any combination, how many different combinations are possible?
I am aware that when only 2x1 dominoes are used then the cell definition using dynamic programming is: $$ f\left [ 0 \right ]= 0 $$ $$ f\left [ 1 \right ]= 1, i=1 $$ $$ f\left [ 2 \right ]= 2, i=2 $$ $$ f\left [ i \right ]= f\left [ i-2 \right ]+ f\left [ i-1 \right ], \forall i>2 $$
When L shaped trominoes are added to consideration, we (assume) can use them only in pairs since using odd number L shaped trominoes will leave one 1x1 grid unfilled. so when a pair of such trominoes combine it is equivalent to using a single 3x2 rectangular piece. Everytime we use $\ num=\lfloor n/3 \rfloor $ rectangular pieces and the rest are filled with 2x1 dominoes. Also the total number of combinations would be sum of twice the $\ num$, $\ f[i] $ and number of 2x1 domino combinations of the rest of the grid when the 3x2 piece is used. Is this assumption right? This is how far I have gotten through. If any of you have any hints it would be appreciated. Thank You in advance.
Note: Tile Rotations are allowed