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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). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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).
LINKS
FORMULA
a(n) = A000002(ceiling(2*n/3)) (conjectured). - Jon Maiga, Jan 24 2021
EXAMPLE
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, ...
PROG
(PARI) See Links section.
CROSSREFS
Sequence in context: A246465 A172497 A211226 * A135265 A144110 A076490
KEYWORD
nonn,easy
AUTHOR
Rémy Sigrist, Feb 10 2019
STATUS
approved

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Last modified March 29 05:43 EDT 2024. Contains 371264 sequences. (Running on oeis4.)