For this problem looking for a pattern in the units digits would help if we were dividing by $10$, as the remainder of a number when we divide it by $10$ is precisely its units digit. 

Try doing the same but looking at the remainders when you divide by $11$, that is, if you find a number in the sequence that is $11$ or more, write its remainder when dividing by $11$ and use that as the number in the sequence. So, instead of your sequence looking like $1,1,3,5,8,13,21\dots$, it would look like $1,1,2,3,5,8,2,10,\dots$