Tray Bien

This problem asks us to fill a 3-by-x foot space with trays of 2x1 or 1x1 and report back the number of ways to do it.

This is a pretty standard DP problem. The idea is that we systematically fill from the left and track how many ways the "front' of trays can look. It turns out there are 7:

# # # X # # # X # # # X X X # X # X # X X X X X