A puzzle

I have another idea for a longer proof-post with nice pictures, but it’s going to take a bit of time to prepare.  Meanwhile, here’s a quick and dirty puzzle.

A sequence begins as follows:

1

11

21

1211

111221

What’s the pattern here?  Finding it takes cleverness, but absolutely no sophisticated math.

Below this heart-melting video of Bill Evans’ trio in 1970, I’ll put down one more term as a big extra hint.

Highlight this decoy list of numbers to see the hint (this won’t work on your dashboard): 121313231221122321