A288725 - OEIS

%I #23 Jan 17 2019 22:51:14

%S 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,

%T 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,

%U 3,1,1,2,2,3,3,1,1,1,2,2,2,3,3,3,1,2,2

%N 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).

%C See comments at A288723.

%H Georg Fischer, <a href="/A288725/b288725.txt">Table of n, a(n) for n = 1..2000</a> (recovered b-file, Jan 16 2019)

%e 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:

%e (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, ...

%e Apply the same process to the second sequence and the third sequence appears:

%e (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, ...

%e Apply the same process to the third sequence and the original sequence reappears:

%e (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, ...

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

%K nonn

%O 1,1

%A _Anthony Sand_, Jun 14 2017