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A078880 The sequence starting with 2 that equals its own run length sequence.

%I A078880 #17 Nov 30 2016 12:33:22
%S A078880 2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1,2,
%T A078880 2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,
%U A078880 1,1,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1
%N A078880 The sequence starting with 2 that equals its own run length sequence.
%C A078880 It is an unsolved problem to show that the density of 1's is equal to 1/2.
%C A078880 The sequence can be generated by starting with 22 and applying the block-substitution rules 22 -> 2211, 21 -> 221, 12 -> 211, 11 -> 21. (Lagarias)
%D A078880 M. S. Keane, Ergodic theory and subshifts of finite type, Chap. 2 of T. Bedford et al., eds., Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford, 1991, esp. p. 50.
%H A078880 Ivan Neretin, <a href="/A078880/b078880.txt">Table of n, a(n) for n = 1..10000</a>
%H A078880 J.-M. Fedou, G. Fici, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Fici/fici.html">Some remarks on differentiable sequences and recursivity</a>, JIS 13 (2010) # 10.3.2.
%F A078880 a(n) = k(n+1), where k=A000002, the Kolakoski sequence.
%e A078880 Start with 2, which generates 22 (so that the first run length is 2); then 22 generates 2211 (so that the first two run lengths are 2 and 2); then 2211 generates 221121 and so on.
%t A078880 seed = {2, 1}; w = {}; i = 1; Do[w = Join[w, Array[seed[[Mod[i - 1, Length[seed]] + 1]] &, If[i > Length[w], seed, w][[i]]]]; i++, {n, 70}]; w (* _Ivan Neretin_, Apr 02 2015 *)
%Y A078880 See A000002, this sequence prepended with 1, for properties, formulas, references, links, programs, etc.
%K A078880 nonn
%O A078880 1,1
%A A078880 _Clark Kimberling_, Dec 11 2002