Continue the series

A certain thing can be done with the list 2, 2, 1, 1, 0 to obtain the sequence
2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 1, 0, 0, 2, 2, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 2, 1, 1, 1, 1, 0, 0, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 0, 2, 2, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 0, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 0, 2, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 1, 0, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 0, 0, 2, 2, 1, 0, 0, 2, 2, 1, 0, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 0, 2, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 0, 0, 2, 2, 2, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 2, 2, 1, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 1, 0, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 0, 2, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 1, 1, 0, 2, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 0, 2, 2, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 1, 0, 0, 2, 2, 2, 1, 1, 1, 1, 0, 0, 2, 2, 1, 0, 0, 2, 2, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 2, 1, 1, 0, 2, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 0, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 0, 2, 2, 0, 0, 2, 2, 2, 0, 0, 2, 2, 2, 1, 0, 0, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 2, 1, 1, 1, 1, 0, 0, 2, 2, 1, 1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 0, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 2, 1, 1, 0, 0, 2, 2, 1, 1, 1, 1, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 2, 2, 2, 1, 1, 1, 1, 0, 0, 2, 2, 2, 1, 1, 0, 2, 2, 1, 1, 1, 1, 0, 2, 2, 1, 2, 2, 2, 2, 1, 1, 0, 2, 1, 1, 0, 2, 2, 1, 0, 0, 2, 2, 2, 2, 1, 1, 1, 0, 2, 2, 1, 1, 0, 0, 2, 2, 0, 2, 2, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 1, 0, 2, 2, 0, 0, 2, 2, 2, 2, 1, 1, 0, 1, 1, 1, 1, 0, 0, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 0, 2, 2, 1, 0, 0, 2, 2, 2, 1, 1, 1, 1, 0, 0, 2, 2, 2, 1, 0, 0, 2, 2, 1, 1, 0, 0, 2, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 0, 2, 2, 1, 1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 0, 0, 2, 2, 1, 1, 1, 1, 0, 2, 2, 2, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 0, 0, 2, 2, 1, 0, 2, 2, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 2, 2, 1, 1, 2, 2, 2, 1, 1, 0, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 0, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 2, 2, 2, 1, 0, 0, 2, 2, 2, 1, 1, 2, 2, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 1, 0, 2, 1, 1, 0, 0, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 0, 2, 1, 1, 1, 0, 2, 2, 1, 1, 1, 1, 0, 2, 2, 1, 2, 2, 2, 2, 1, 0, 0, 2, 2, 1, 1, 0, 0, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 0, 2, 2, 2, 2, 1, 1, 1, 0, 2, 2, 1, 1, 1, 0, 2, 2, 2, 1, 1, 1, 0, 2, 2, 1, 1, 0, 0, 2, 2, 2, 1, 1, 1, 1, 0, 0, 2, 2, 1, 1, 1, 0, 0, 2, 2, 1, 0, 2, 2, 1, 0, 2, 2, 0, 0, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1, 0, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 1, 1, 0, 0, 2, 2, 1, 0, 0, 2, 2, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 2, 2, 2, 2, 1, 1, 0, 2, 2, 1, 1, 0, 2, 2, 1, 1, 1, 0, 2, 2, 1, 1, 0, 0, 2, 2, 2, 1, 1, 1, 1, 0, 0, 2, 1, 1, 0, 0, 2, 2, 0, 0, 2, 2, 2, 2, 1, 1, 0, 0, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 0, 2, 2, 1, 1, 0, 0, 2, 2, 1, 0, 0, 2, 2, 2, 2, 1, 1, 0, 2, 2, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 2, 2, 1, 1, 2, 2, 1, 1, 1, 0, 2, 1, 1, 0, 0, 2, 1, … (It is higly likely that this sequence continues endlessly)

The same thing, when done with the list 3, 1, 2, 0, 0, yields the sequence
3, 3, 3, 1, 1, 1, 2, 2, 2, 0, 0, 3, 1, 1, 2, 2, 0, 0, 1, 1, 1, 2, 0, 0, 0, 3, 3, 0, 0, 3, 1, 1, 3, 3, 3, 1, 1, 1, 0, 0, 0, 3, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 3, 1, 2, 1, 1, 1, 2, 0, 0, 0, 0, 3, 3, 2, 2, 2, 0, 0, 0, 3, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 1, 1, 2, 2, 1, 1, 1, 2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 3, 1, 1, 2, 2, 0, 0, 3, 1, 1, 0, 0, 3, 3, 1, 1, 1, 1, 1, 2, 0, 0, 0, 3, 3, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 3, 1, 2, 0, 0, 3, 3, 0, 0, 0, 3, 3, 3, 3, 3, 3, 2, 2, 2, 0, 0, 0, 2, 2, 2, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0, 0, 3, 3, 3, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 3, 3, 0, 0, 3, 3, 1, 1, 3, 3, 1, 1, 2, 2, 0, 0, 0, 3, 3, 3, 1, 1, 1, 2, 0, 0, 3, 3, 1, 1, 2, 2, 2, 2, 2, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 3, 3, 3, 1, 1, 1, 2, 0, 0, 0, 3, 3, 0, 0, 0, 3, 3, 3, 1, 1, 1, 2, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 3, 1, 2, 2, 0, 0, 0, 0, 3, 3, 1, 1, 3, 1, 2, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 3, 3, 3, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 1, 2, 0, 0, 0, 2, 2, 2, 0, 0, 0, 3, 3, 3, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 3, 3, 3, 2, 2, 2, 0, 0, 0, 0, 0, 0, 3, 1, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 3, 1, 1, 3, 3, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 3, 3, 1, 1, 3, 3, 3, 1, 2, 2, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 0, 3, 3, 1, 1, 0, 0, 0, 0, 3, 1, 1, 2, 2, 0, 0, 3, 3, 3, 1, 2, 0, 0, 0, 0, 0, 0, 3, 1, 2, 2, 0, 0, 0, 3, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 1, 2, 0, 0, 0, 0, 0, 0, 3, 3, 3, 1, 2, 2, 0, 0, 0, 0, 3, 1, 1, 2, 2, 0, 0, 0, 0, 0, 3, 3, 3, 1, 2, 2, 2, 2, 0, 0, 3, 3, 1, 1, 0, 0, 0, 3, 3, 3, 1, 1, 1, 2, 0, 0, 3, 3, 3, 1, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 3, 3, 3, 1, 2, 2, 0, 0, 0, 3, 3, 3, 1, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 3, 1, 2, 2, 0, 0, 0, 0, 0, 3, 3, 3, 1, 1, 1, 2, 0, 0, 0, 0, 3, 3, 1, 1, 0, 0, 0, 3, 3, 3, 1, 2, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 3, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 2, 2, 0, 0, 0, 3, 3, 0, 0, 0, 0, 3, 3, 0, 0, 0, 3, 3, 3, 2, 2, 2, 0, 0, 0, 0, 0, 0, 3, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 3, 1, 2, 2, 3, 3, 3, 1, 2, 2, 0, 0, 0, 0, 0, 3, 3, 3, 1, 1, 1, 2, 0, 0, 3, 3, 3, 3, 3, 1, 1, 1, 2, 0, 2, 2, 2, 0, 0, 0, 0, 0, 0, 3, 1, 1, 2, 0, 0, 3, 3, 1, 1, 2, 2, 2, 2, 2, 0, 0, 0, 3, 3, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 3, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 3, 3, 0, 0, 0, 3, 3, 3, 1, 1, 1, 2, 2, 0, 0, 0, 0, 1, 1, 1, 2, 0, 0, 0, 3, 3, 1, 2, 0, 0, 0, 3, 3, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 3, 3, 3, 1, 1, 1, 2, 2, 2, 0, 0, 0, 3, 3, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 3, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 3, 1, 2, 0, 0, 3, 3, 1, 1, 2, 2, 0, 0, 0, 3, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 0, 0, 0, 3, 3, 1, 1, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 3, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 3, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0, 0, 3, 1, 2, 0, 0, 0, 0, 0, 3, 1, 2, 2, 1, 1, 1, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 2, 0, 0, 3, 3, 3, 3, 3, 1, 1, 1, 2, 2, 2, 0, 0, 3, 1, 1, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 3, 1, 2, 2, 0, 0, 0, 0, 0, 0, 0, 3, 1, 1, 2, 2, 0, 0, 0, 0, 0, 0, 3, 3, 3, 1, …

What is the rule for forming these long sequences from the short lists? Can you continue the sequences?

Hints:

• It is actually quite easy. You don't even need the four basic arithmetic operations, let alone something more complicated.
• It is actually quite easy. Nonetheless I find it difficult to produce more than the first 20 sequence terms in my head.
• It is actually quite easy. It would be even easier if I told you what becomes of an input list that does not contain zeros or repetitions.
• It is actually quite easy. I was able to express the rule as a Haskell program of just 59 bytes length. Sacrificing maximum efficiency, it is even possible with 52 bytes. 52 bytes is this long:
  ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz