Look and say

A simple look and say sequence is as follows:

1

11

21

1211

111221

311211

13211221

This is read as 1, one 1, two 1, etc. This sequence is interesting because you will never get a 4 unless the starting number is. This is because each line can be divided into a data sequence of the form xN (where x is how many and N is the number described). Now the best way to show there will not be a 4 is to work backwards. So we would have to start with somthing like this:

41

Now to get that we would have to have on the preceding line:

1111

41

and on the previous line:

11

1111

411

which is impossible since that would be one one and one one which is not a legal construction.
Intriguing, huh?