Hints for recursive sequence problem?

Question

I am working on this problem:

I would just like a hint if possible. This is what I've done so far:

$1)$ Calculated the sequence to more terms (if I didn't make a mistake): $2, 7, 1, 4, 7, 4, 2, 8, 2, 8, 8, 1, 6, 1, 6, 1, 6, 6,4, 8, 6, 6, 6, 6, 6, 3, 6, 2, 4, 3, 2.$

$2)$ I wrote the positions of the first few $6's$: $13, 15, 17, 18, 21,22,23,24,25,27,35,37, 39$

I couldn't find a pattern anywhere.

EDIT: I eventually arrived at the same approach that Anatoly gave in his answer (that is, looking for a self-reproducing chunk). However, I couldn't find any such chunk. I am wondering what is the best way to find such a chunk?

Answer 1

We have to look for a pattern of digits that occur in a cyclic manner and that contain the digit $6$. For example, if the sequence $888$ occurs, it begets as successive sequences $6464 \,$, $2424...\, $, $888\,$. So if we demonstrate that a triple $888$ occurs, it necessarily implies that there are infinite $6$ in the whole sequence.

Actually the triple $888\,$ occurs: you have already noted that the sequence contains $636\,$, which begets $1818\,$, which in turn begets $888\,$.