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A284388 0-limiting word of the morphism 0 -> 1, 1 -> 001. 4

%I #40 Sep 14 2023 00:51:40

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

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

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

%N 0-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 Conjecture: after the first term, same as one less than the run-lengths of the Thue-Morse sequence A010060. - _George Beck_, Mar 24 2021

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

%C Conjecture: Excluding the first two terms, 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 two terms, 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="/A284388/b284388.txt">Table of n, a(n) for n = 1..10000</a>

%t s = Nest[Flatten[# /. {0 -> {1}, 1 -> {0, 0, 1}}] &, {0}, 8]; (* this sequence *)

%t Flatten[Position[s, 0]]; (* A284389 *)

%t Flatten[Position[s, 1]]; (* A284390 *)

%Y Cf. A010060, A284389, A284390, A284391, A003159, A026465.

%K nonn,easy

%O 1

%A _Clark Kimberling_, Mar 30 2017

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Last modified March 28 08:22 EDT 2024. Contains 371236 sequences. (Running on oeis4.)