A real Challenge!

[no genius]

can you give the next two rows?

1
11
21
1211
111221
312211
13112221
1113213211
31131211131221
-------------------
-------------------

[1+1=1]

maybe the next row is 12212311321211?

[jamesrc]

The next rows are:

13211311123113112211
and then
11131221133112132113212221
(each line tells you what was in the previous line)
but this is not number theory; it should be in the brain teaser section.

[Gokul43201]

Also, it's been done already in the Brain Teaser subforum...but is an fun puzzle, nevertheless - requires some out-of-the-box thinking.