[seqfan] Replacing chunks by their average

[Eric Angelini]

S=
1,2,2,2,1,3,2,2,1,2,4,3,1,2,1,2,1,7,2,4,1,2,1,2,1,2,12,3,1,4,1,2,1,2,1,2,1,23,
2,4,1,4,1,2,1,2,1,2,1,2,44,3,14,14,1,2,1,2,1,2,1,2,1,87,2,4,1,4,1,4,1,2,1,2,1,
2,1,2,1,2,172,3,1,4,1,4,1,4,1,2,1,2,1,2,1,2,1,2,1,343,...

Hello SeqFans,
Replace in S all chunks of same-parity integers by the average of the said chunk and you will get S back.

Example:
- the first chunk is made by 1, -- replace it with 1;
- the second chunk is 2,2,2, -- replace it with 2;
- the third chunk is 1,3, -- replace it with 2;
- the fourth chunk is 2,2, -- replace it with 2;
- the fifth chunk is made by 1, -- replace it with 1;
- the sixth chunk is made by 2,4, -- replace it by 3;
etc.
The replacements so far have been 1,2,2,2,3,1... which rebuilds S.

----
In considering S, one quickly notices this array where the rightmost diagonal is A023105:

1,
2,
2,
2,1,3,
2,2,1,2,4,
3,1,2,1,2,1,7,
2,4,1,2,1,2,1,2,12,
3,1,4,1,2,1,2,1,2,1,23,
2,4,1,4,1,2,1,2,1,2,1,2,44,
3,14,14,1,2,1,2,1,2,1,2,1,87,
2,4,1,4,1,4,1,2,1,2,1,2,1,2,1,2,172,
3,1,4,1,4,1,4,1,2,1,2,1,2,1,2,1,2,1,343,
...

Best,
É.