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{001->2}-transform of the infinite Fibonacci word A003849.
7

%I #28 Mar 28 2022 21:38:35

%S 0,1,2,0,1,2,2,0,1,2,0,1,2,2,0,1,2,2,0,1,2,0,1,2,2,0,1,2,0,1,2,2,0,1,

%T 2,2,0,1,2,0,1,2,2,0,1,2,2,0,1,2,0,1,2,2,0,1,2,0,1,2,2,0,1,2,2,0,1,2,

%U 0,1,2,2,0,1,2,0,1,2,2,0,1,2,2,0,1,2

%N {001->2}-transform of the infinite Fibonacci word A003849.

%C From _Dan Rust_, Aug 18 2018: (Start)

%C This word is the fixed point of the morphism 0->2, 1->01, 2->2201 with the finite string 0, 1, 2, 0, 1 appended to the beginning. This morphism comes from taking the 'proper' version of the Fibonacci morphism 0->01, 1->1, given by 0->001, 1->01 (A189661 but with the rightmost 0 moved to the left of each image word), then replacing 001 with 2 and noting that the new symbol 2 should map to 00100101 = 2201 in order to be consistent.

%C The finite string appended to the beginning comes from the process of finding a proper version of the Fibonacci morphism using a return word encoding and taking conjugates which causes a shift of the respective fixed points.

%C (End)

%C This sequence is the unique fixed point of the morphism 0->01, 1->2, 2->0122. See the paragraph following Lemma 23 in the paper by Allouche and me. - _Michel Dekking_, Oct 05 2018

%H Clark Kimberling, <a href="/A284749/b284749.txt">Table of n, a(n) for n = 1..10000</a>

%H J.-P. Allouche, F. M. Dekking, <a href="https://arxiv.org/abs/1809.03424">Generalized Beatty sequences and complementary triples</a>, arXiv:1809.03424v3 [math.NT], 2018-2019.

%e As a word, A003849 = 01001010010010100..., and replacing each 001 by 2 gives 01201220120...

%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13] (* A003849 *)

%t w = StringJoin[Map[ToString, s]]

%t w1 = StringReplace[w, {"001" -> "2"}]

%t st = ToCharacterCode[w1] - 48 (* A284749 *)

%t Flatten[Position[st, 0]] (* A214971 *)

%t Flatten[Position[st, 1]] (* A284624 *)

%t Flatten[Position[st, 2]] (* A284625 *)

%Y Cf. A003849, A214971, A284624, A284625.

%K nonn,easy

%O 1,3

%A _Clark Kimberling_, May 02 2017