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1-limiting word of the morphism 0 -> 1, 1 -> 001.
4

%I #48 Jan 14 2025 10:21:04

%S 1,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,1,1,0,0,1,1,1,0,0,1,0,0,1,

%T 0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,1,1,0,0,

%U 1,1,1,0,0,1,0,0,1,0,0,1,1,1,0,0,1,1

%N 1-limiting word of the morphism 0 -> 1, 1 -> 001.

%C The morphism 0 -> 1, 1 -> 001 has two limiting words. If the number of iterations is even, the 0-word evolves from 0 -> 1 -> 001 -> 11001 -> 00100111001; if the number of iterations is odd, the 1-word evolves from 0 -> 1 -> 001 -> 11001 -> 00100111001 -> 110011100100100111001, as in A284391. The 0-limiting word results from the 1 limiting word by replacing the initial 00 by 1.

%C Conjecture: the limiting frequency of 0's in both limiting words is 1/2.

%C From _Adnan Baysal_, Aug 06 2023: (Start)

%C Conjecture: Excluding the first term, if runs of 1s of length one are replaced by 0, and runs of 1s of length three are replaced by 1, we get the same sequence.

%C Conjecture: Index distance between two consecutive 1s is either one or three. Excluding the first term, if distances of one are replaced by 0, and distances of three are replaced by 1, we get the same sequence. (End)

%H Clark Kimberling, <a href="/A284391/b284391.txt">Table of n, a(n) for n = 1..10000</a>

%H Jean-Paul Allouche, <a href="https://webusers.imj-prg.fr/~jean-paul.allouche/121-12221.pdf">On the morphism 1 -> 121, 2 -> 12221</a>, CNRS France, 2024. See pp. 7-9.

%H Jean-Paul Allouche, <a href="/A026465/a026465.pdf">On the morphism 1 -> 121, 2 -> 12221</a>, Preprint, 2024 [Local copy, with permission]

%t s = Nest[Flatten[# /. {0 -> {1}, 1 -> {0, 0, 1}}] &, {0}, 9]; (* this sequence *)

%t Flatten[Position[s, 0]]; (* A284392 *)

%t Flatten[Position[s, 1]]; (* A284393 *)

%Y A284388 shifted right.

%Y Cf. A003159, A284392, A284393.

%K nonn,easy

%O 1

%A _Clark Kimberling_, Mar 30 2017