Tiling with L's and squares
We consider tilings of 2 × n, 3 × n, and 4 × n boards with 1 × 1 squares and L-shaped tiles covering an area of three square units, which can be used in four different orientations. For the 2 × n board, the recurrence relation for the number of tilings is of order three and, unlike most third order recurrence relations, can be solved exactly. For the 3 × n and 4 × n board, we develop an algorithm that recursively creates the basic blocks (tilings that cannot be split vertically into smaller rectangular tilings) of size 3 × k and 4 × k from which we obtain the generating function for the total number of tilings. We also count the number of L-shaped tiles and 1 × 1 squares in all the tilings of the 2 × n and 3 × n boards and determine which type of tile is dominant in the long run.