I couldn't figure this one out, but saw the solution. Below is a hint, followed by a start.
Hint: look at all the possible states the blocks could be in as the stairs are built, and the ways to reach those states.
On the first row of a sheet of paper, draw all (one) ways the first block can be placed.
On the second row, draw both possible ways two blocks can be placed, and draw lines between the first row image and these.
On the third row, draw all three possible states for the blocks, and connect them with the states from the previous row from which they might have come. At this point you need to start writing in the counts-1 for the cases of three on the floor and three stacked, and 2 for the case of stairs.
On each subsequent row, the number of routes to a state is the sum of the numbers for each prior row state that could have led to it.
Be careful! I missed two of the seven possible states for six blocks and came up with an answer that was less than 1/2 of the correct answer.