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The word 1 belongs to the sequence, and whenever a word w belongs to the sequence, then the words consisting of 1's and 2's whose run lengths transform equals w also belong to the sequence.
8

%I #19 Oct 03 2024 11:02:26

%S 1,2,11,12,21,22,112,122,211,221,1121,1122,1211,2122,2211,2212,11221,

%T 12112,12211,12212,21121,21122,21221,22112,112212,121122,212211,

%U 221121,1121122,1121221,1122122,1211221,1221121,1221211,2112122,2112212,2122112,2211211

%N The word 1 belongs to the sequence, and whenever a word w belongs to the sequence, then the words consisting of 1's and 2's whose run lengths transform equals w also belong to the sequence.

%C This sequence lists finite smooth words: finite words w composed of 1's and 2's without three or more consecutive equal digits, such that for any k > 0, the k-th iterate of the run lengths transform of w is also a word composed of 1's and 2's without three or more consecutive equal digits.

%H Geneviève Paquin, Srĕcko Brlek, Damien Jamet, <a href="https://www.irisa.fr/JM06/Slides/SlidesPaquinBrlek.pdf">Extremal generalized smooth words</a>

%H Rémy Sigrist, <a href="/A376637/b376637.txt">Table of n, a(n) for n = 1..10048</a>

%H Rémy Sigrist, <a href="/A376637/a376637.png">Illustration of the first terms</a> (arrows denotes run lengths transforms)

%H Rémy Sigrist, <a href="/A376637/a376637.gp.txt">PARI program</a>

%H <a href="/index/K#Kolakoski">Index entries for sequences related to Kolakoski sequence</a>

%e The first terms, alongside their run lengths transform, are:

%e n a(n) RL(a(n))

%e -- ---- --------

%e 1 1 1

%e 2 2 1

%e 3 11 2

%e 4 12 11

%e 5 21 11

%e 6 22 2

%e 7 112 21

%e 8 122 12

%e 9 211 12

%e 10 221 21

%e 11 1121 211

%e 12 1122 22

%e 13 1211 112

%e 14 2122 112

%e 15 2211 22

%e 16 2212 211

%o (PARI) \\ See Links section.

%Y Cf. A000002, A007931, A376638, A376674, A376676, A376685, A376698.

%K nonn,base

%O 1,2

%A _Rémy Sigrist_, Sep 30 2024