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A306366 For any sequence s of positive integers without infinitely many consecutive equal terms, let T(s) be the sequence obtained by replacing each run, say of k consecutive t's, with a run of t consecutive k's; this sequence corresponds to T(K) (where K denotes the Kolakoski sequence A000002).

%I A306366 #19 Feb 15 2021 02:02:47
%S A306366 1,2,2,2,1,1,1,2,2,1,2,2,2,1,1,2,2,2,1,1,1,2,1,1,1,2,2,2,1,1,2,1,1,1,
%T A306366 2,2,1,2,2,2,1,1,1,2,2,1,1,1,2,1,1,2,2,2,1,2,2,2,1,1,1,2,2,1,2,2,2,1,
%U A306366 1,2,1,1,1,2,2,1,1,1,2,2,2,1,2,2,2,1,1
%N A306366 For any sequence s of positive integers without infinitely many consecutive equal terms, let T(s) be the sequence obtained by replacing each run, say of k consecutive t's, with a run of t consecutive k's; this sequence corresponds to T(K) (where K denotes the Kolakoski sequence A000002).
%C A306366 If s is finite, then s and T(s) have the same sum.
%C A306366 Fixed points of T correspond to sequences where each run, say of t's, has t elements; A001650, A001670, A002024, A130196, A167817, A175944 and A213083 are fixed points of T.
%C A306366 When s has no consecutive equal terms, then T(s) is all 1's (A000012).
%C A306366 Apparently, T^4(K) = T^2(K) (where T^i denotes the i-th iterate of K).
%H A306366 Rémy Sigrist, <a href="/A306366/a306366.gp.txt">PARI program for A306366</a>
%F A306366 a(n) = A000002(ceiling(2*n/3)) (conjectured). - _Jon Maiga_, Jan 24 2021
%e A306366 The first terms of the Kolakoski sequence are:
%e A306366        +-----+     +--+  +-----+  +-----+     +--
%e A306366        |     |     |  |  |     |  |     |     |
%e A306366     +--+     +-----+  +--+     +--+     +-----+
%e A306366     |#1|#2   |#3   |#4|#5|#6   |#7|#8   |#9   |#10 ...
%e A306366     +--+-----+-----+--+--+-----+--+-----+-----+--
%e A306366       1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, ...
%e A306366 .
%e A306366 The first terms of this sequence are:
%e A306366        +-----+--+        +-----+  +-----+--
%e A306366        |     .  |        |     |  |     .
%e A306366     +--+     .  +-----+--+     +--+     .
%e A306366     |#1|#2   .#3|#4   .#5|#6   |#7|#8   .#9  ...
%e A306366     +--+-----+--+-----+--+-----+--+-----+--
%e A306366       1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, ...
%o A306366 (PARI) See Links section.
%Y A306366 Cf. A000002, A000012, A001650, A001670, A002024, A130196, A167817, A175944, A213083.
%K A306366 nonn,easy
%O A306366 1,2
%A A306366 _Rémy Sigrist_, Feb 10 2019