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)
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:
ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz