A213083 -id:A213083 - cp's OEIS frontend

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).

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, 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, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 2, 1, 1

Offset: 1

Rémy Sigrist, Feb 10 2019

If s is finite, then s and T(s) have the same sum.
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.
When s has no consecutive equal terms, then T(s) is all 1's (A000012).
Apparently, T^4(K) = T^2(K) (where T^i denotes the i-th iterate of K).

			The first terms of the Kolakoski sequence are:
       +-----+     +--+  +-----+  +-----+     +--
       |     |     |  |  |     |  |     |     |
    +--+     +-----+  +--+     +--+     +-----+
    |#1|#2   |#3   |#4|#5|#6   |#7|#8   |#9   |#10 ...
    +--+-----+-----+--+--+-----+--+-----+-----+--
      1, 2, 2, 1, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, ...
.
The first terms of this sequence are:
       +-----+--+        +-----+  +-----+--
       |     .  |        |     |  |     .
    +--+     .  +-----+--+     +--+     .
    |#1|#2   .#3|#4   .#5|#6   |#7|#8   .#9  ...
    +--+-----+--+-----+--+-----+--+-----+--
      1, 2, 2, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, ...

Rémy Sigrist, PARI program for A306366

Cf. A000002, A000012, A001650, A001670, A002024, A130196, A167817, A175944, A213083.

PARI
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See Links section.
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a(n) = A000002(ceiling(2*n/3)) (conjectured). - Jon Maiga, Jan 24 2021

cp's OEIS Frontend

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).

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