1

11

21

1211

111221

312211

13112221

1113213211

31131211131221

13211311123113112211

11131221133112132113212221

3113112221232112111312211312113211

1321132132111213122112311311222113111221131221

*etc.*
The way this sequence works is described in 1, 11, 21, 1211.

Things that are interesting to prove about this sequence, in order of difficulty:

- There will never be a 4, or anything higher, in this sequence.
- If the sequence started with 2 instead of 1, every number after the 2nd would end with '112'.
- The sequence will never get shorter.
- If you consider a new sequence, comprised of digits
*m* through *n* of each step, this sequence will eventually cycle. (For example, this sequence where m=1 and n=2 is 11, 21, 12, 11, 31, 13, 11, 31, 13 ...)