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A249094 Length of reverse self-iteration of the Kolakoski sequence A000002 starting at A000002(n): a(n) = max { k | A000002(n-i+1) = A000002(i), 0 < i <= k }. 6
0, 0, 4, 1, 0, 2, 0, 0, 4, 0, 0, 4, 1, 0, 2, 1, 0, 0, 7, 0, 2, 1, 0, 2, 0, 0, 4, 1, 0, 2, 1, 0, 2, 0, 0, 4, 0, 0, 4, 1, 0, 2, 0, 0, 4, 0, 2, 1, 0, 2, 1, 0, 0, 7, 0, 0, 4, 1, 0, 2, 0, 0, 4, 0, 0, 4, 1, 0, 2, 1, 0, 2, 0, 0, 4, 0, 2, 1, 0, 0, 11, 0, 0, 4, 1, 0, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

COMMENTS

The Kolakoski sequence A000002 has a fractal structure that appears in the infinite number of iterations and reverse iterations of itself that it contains. Each iteration develops itself in two branches, a right branch in the direct sense, and a left branch in the reverse sense, e.g., 122-1-221121. This sequence gives the length of the reverse iteration (or left branch) starting at position n, with a length = 0 if A000002(n) = 2 <> A000002(1) = 1.

The lengths of the right branches are in A249093 and the lengths of the full iterations with the two branches are in A249507.

Recalling that A000002 begins as 1221121221..., the apparition of these iterations is easily understood from the evolution of an initial 2 in even position in A000002, which generates: 2 > (1)22(1) > (2)122112(1) > (1)221221121221(2)... (as long as the equivalent of the initial 2 in the successive iterates remains in even position).

Because each iteration must be generated by a preceding (and shorter) iteration, each branch is constituted of a term of A054351 (successive generations of the Kolakoski sequence) in reverse order for the left branches, and the nonzero values of this sequence are all in A054352. Any given value > 1 cannot appear in this sequence before the other smaller values.

LINKS

Jean-Christophe Hervé, Table of n, a(n) for n = 2..99990

EXAMPLE

A000002(n) = 2 => a(n) = 0 since the Kolakoski sequence begins with 1. a(10) = 4 since A000002(7:10) = A000002(1:4) and A000002(6) <> A000002(5).

CROSSREFS

Cf. A000002, A054351, A054352, A249093, A249507, A249508.

Sequence in context: A058710 A281891 A124539 * A096501 A062862 A206799

Adjacent sequences:  A249091 A249092 A249093 * A249095 A249096 A249097

KEYWORD

nonn

AUTHOR

Jean-Christophe Hervé, Oct 30 2014

STATUS

approved

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Last modified September 18 07:58 EDT 2019. Contains 327168 sequences. (Running on oeis4.)