Define a sequence a(n) such that a(3k)=0, a(3k+1)=1, and a(3k+2) = a(k) for all non-negative integer k.
The sequence starts out as: 0 1 0 0 1 1 0 1 0 0 1 0 0 1 ...

Prove there is no string that repeats three consecutive times. For example 1 0 1 0 1 0 will never occur (1 0 three times). Note that 1 0 1 0 1 1 0 can occur (the extra one breaks the repetition).