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%I #37 Jan 17 2019 22:50:27
%S 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,
%T 3,1,2,3,1,1,2,2,2,3,1,2,2,3,3,1,1,1,2,2,2,3,3,3,1,2,3,1,1,2,2,3,3,1,
%U 1,1,2,3,3,1,1,1,2,2,2,3,1,2,2,3,3,1,1
%N Second sequence of a Kolakoski 3-Ouroboros, i.e., sequence of 1s, 2s and 3s that is second 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="/A288724/b288724.txt">Table of n, a(n) for n = 1..2000</a> (recovered b-file, Jan 16 2019)
%H Rémy Sigrist, <a href="/A288724/a288724.gp.txt">PARI program for A288723, A288724 and A288725</a>
%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 (as above). The run-lengths of this second sequence yield a third and different sequence, A288725. 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, ...
%o (PARI) See Links section.
%Y Cf. A000002, A025142, A025143. A288723 and A288725 are the first and third sequences in this 3-Ouroboros.
%K nonn
%O 1,1
%A _Anthony Sand_, Jun 14 2017
%E Data corrected by _Rémy Sigrist_, Oct 07 2017
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