Alright. Does each puzzle appear to be solved and acknowledged? (plane, and 100 meter stick)

I have the following one, and for once, it's not one you will find on every riddle site. I knew the question for a while, and worked out the (surprisingly easy) proof today after giving it a proper shot, so it should be doable. I have no clue how easy this is for people who aren't used to tackling math problems, so I will give hints generously if necessary.

Consider the sequence (a

_{n}) for n = 0, 1, 2, 3, 4, ..., with possible values only 0 and 1 for each a

_{n}, constructed as follows:

- If n = 3k for some integer k, then a
_{n} = 0. - If n = 3k + 1 for an integer k, then a
_{n} = 1. - If n = 3k + 2 for an integer k, then a
_{n} = a_{k}.

Thus, the sequence starts as follows: 0 1

**0** 0 1

**1** 0 1

**0** 0 1

**0** 0 1 ... The bold digits come from the third rule, there's nothing special to them. It's just to visualize the rule.

We define a

*triple* to be any non-empty substring of the sequence (a

_{n}) together with two repetitions of itself immediately following it. E.g. 101010 somewhere in the sequence is a triple, but 0010 is not, since the second repetition of 0 isn't immediately after the 00.

Show that there are no triples in the sequence (a

_{n}).

-- Simon