Tile a board with two tile types: dominoes (horizontal or vertical) and L-shaped trominoes (each covering cells). Count the number of tilings modulo , where can be up to . Trominoes add new profile states: you can have one row sticking out one cell further than the other.
This creates profiles you didn't see with dominoes alone. With up to , you need an time, space solution. Broken profile DP with rows gives you states and formulas that lead to a recurrence.