A249949 Number of generations between each single 1 in the Kolakoski sequence A000002 and its nearest double 1 ancestor.

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

Offset: 1

Author

Jean-Christophe Hervé, Nov 09 2014

Keywords

• nonn

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.