A288725 Third sequence of a Kolakoski 3-Ouroboros, i.e., sequence of 1s, 2s and 3s that is third in a chain of three distinct sequences where successive run-length encodings produce seq(1) -> seq(2) -> seq(3) -> seq(1).

3, 1, 2, 2, 3, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 3, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 3, 3, 1, 1, 2, 2, 3, 3, 3, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 3, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 1, 1, 1, 2, 2, 2, 3, 3, 3, 1, 2, 2

Offset

1,1

Comments

See comments at A288723.

Links

Georg Fischer, Table of n, a(n) for n = 1..2000 (recovered b-file, Jan 16 2019)

Example

Write down the run-lengths of the sequence A288723, or the lengths of the runs of 1s, 2s and 3s. This yields a second and different sequence of 1s, 2s and 3s, A288724. The run-lengths of this second sequence yield a third and different sequence, A288725 (as above). The run-lengths of this third sequence yield the original sequence. For example, bracket the runs of distinct integers, then replace the original digits with the run-lengths to create the second sequence:
(1,1), (2,2), (3,3), (1,1,1), (2), (3), (1,1), (2,2), (3,3,3), (1,1,1), (2,2,2), (3), (1), (2), (3,3), (1,1,1), (2), (3,3), (1,1), (2,2,2), ... -> 2, 2, 2, 3, 1, 1, 2, 2, 3, 3, 3, 1, 1, 1, 2, 3, 1, 2, 2, 3, ...
Apply the same process to the second sequence and the third sequence appears:
(2,2,2), (3), (1,1), (2,2), (3,3,3), (1,1,1), (2), (3), (1), (2,2), (3,3), (1,1), (2,2,2), (3), (1,1), (2,2,2), (3,3,3), (1), (2), (3), ... -> 3, 1, 2, 2, 3, 3, 1, 1, 1, 2, 2, 2, 3, 1, 2, 3, 3, 1, 1, 1, ...
Apply the same process to the third sequence and the original sequence reappears:
(3), (1), (2,2), (3,3), (1,1,1), (2,2,2), (3), (1), (2), (3,3), (1,1,1), (2), (3), (1,1), (2,2), (3,3,3), (1,1,1), (2,2,2), (3), (1), ... -> 1, 1, 2, 2, 3, 3, 1, 1, 1, 2, 3, 1, 1, 2, 2, 3, 3, 3, 1, 1, ...

Crossrefs

Cf. A000002, A025142, A025143. A288723 and A288724 are the first and second sequences in this 3-Ouroboros.

Sequence in context: A072548 A079399 A092155 * A029336 A211939 A010280

Adjacent sequences: A288722 A288723 A288724 * A288726 A288727 A288728

Keyword

nonn

Author

Anthony Sand, Jun 14 2017

Status

approved