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This is an example of the look-and-say sequence, doesn't require so much math

SO for example: when starting with 1, you say 'ONE 1', so the next term would be 11.

You would say 11 as 'TWO one'(as there are two 1s), hence 21.

Therefore following this, the term after '111221' would be 'THREE one TWO two ONE one', so the next term would be '312211'(three 1s, two 2s, and one 1'!

1, 11, 21, 1211, 111221, 1111222111 I guess

Possible sequence identification:

OEIS A001388
1, 11, 21, 1211, 111221, 1012211, 1110112221, 101102110211, 111021101221101221, 1011012211011222110112211, 1110211011222110211022110212221, 10110122110211022110122110222110121110211, ...etc.