PUZZLE #5: WHAT'S THE NEXT NUMBER?

Here are the first few entries of an interesting infinite sequence of numbers:

1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...

The problem is to determine a believable algorithm for generating the rest of the numbers in the sequence, and to specify exactly which (base ten) digits can appear. (solution)