How many ways are there of filling a 2xN board with tetris pieces. I stumbled upon a similar problem and this seemed interesting to solve.
This is my progress: Let S(n) be the number of ways of filling a 2xn board. Evidently, S(2)= 1 and S(4)=4. For n congruent to 2 mod 4, S(n)=2S(n-2)-1 as you can add 2x2 squares to any combination that fills a 2x4 board to either side but you have to subtract one as you would be counting twice to the one composed completely of 2x2 squares. I have been trying to determine S(n) for multiples of 4 but have been unsuccessful. I know I can fill it with ones that can fill 2x4 squares n times, but then for large numbers you can move the 2x2 squares around and these combinations make it hard to determine.