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%I #13 Nov 22 2017 01:15:06
%S 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,
%T 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,
%U 1,1,2,1,2,1,1,2,1,6,1,2,1,1,1,2,1,2,1,4
%N Number of generations between each single 1 in the Kolakoski sequence A000002 and its nearest double 1 ancestor.
%C 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.
%C 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.
%C 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).
%C 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.
%H Jean-Christophe Hervé, <a href="/A249949/b249949.txt">Table of n, a(n) for n = 1..10000</a>
%Y Cf. A000002, A054351, A054352, A249093, A249094, A249507, A249508, A249942, A249948.
%K nonn
%O 1,2
%A _Jean-Christophe Hervé_, Nov 09 2014
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