I am reasonably sure I got the correct count for the one block, one step - 1 way, and the three block, two step - 2 ways solutions. But there are more ways for the six block, three step solutions and then lots more ways for the 10 block, 4 step solutions.
For the six block 3 step solution I came up with these solutions:
BBBGGO, BBBGOG, BBGGOB, BBGOGB, BBGOBG, BGOBGB, BGOBBG
After thinking about this some more I can add BBGGBO, BBGBGO, BBGBOG and I am not sure this is all. About the only rule I can come up with is that in each sequence the number of blocks of the lower level color must always be >= the number of blocks of the next higher level color. So you can't have a sequence that starts out BGG... or BOG... for example.