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A249949 Number of generations between each single 1 in the Kolakoski sequence A000002 and its nearest double 1 ancestor. 3
1, 2, 1, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 4, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 1, 5, 1, 2, 1, 1, 2, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 6, 1, 2, 1, 1, 1, 2, 1, 2, 1, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The single 1's that are considered in this sequence are the 1's between two 2's in the OK sequence A000002 (the first term of A000002 which is indeed a single 1 but not between two 2's is thus not considered here). Each such single 1 is generated by a preceding 1 in the OK sequence that could be single or double, but each single 1 has at least a double 1 in its ancestors since the first 1 of the OK sequence has no descendance except itself. This sequence gives the number of generations between the n-th single 1 in A000002 and its nearest double 1 ancestor. A249942 gives the position of the single 1's in A000002.

The single 1's of the OK sequence are associated with iterated words which develop themselve around each single 1 in two branches; for a description of the iterated words, see comments in A249507 which gives their lengths.

The length of the iterated word around a single 1 is equal to A249507(2*a(n)+1) or to A249507(2*a(n)+2).

Conjecture: this sequence takes all integers k >= 1 as values, so there is no bound to the length of the iterated words that appear in the Kolakoski sequence; the limiting frequency of k is 2/3^k.

LINKS

Jean-Christophe Hervé, Table of n, a(n) for n = 1..10000

CROSSREFS

Cf. A000002, A054351, A054352, A249093, A249094, A249507, A249508, A249942, A249948.

Sequence in context: A276162 A118107 A236548 * A155798 A055652 A290084

Adjacent sequences:  A249946 A249947 A249948 * A249950 A249951 A249952

KEYWORD

nonn

AUTHOR

Jean-Christophe Hervé, Nov 09 2014

STATUS

approved

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Last modified October 26 08:00 EDT 2021. Contains 348267 sequences. (Running on oeis4.)