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A287105 Positions of 0 in A287104.

%I A287105 #21 Oct 03 2019 09:00:40
%S A287105 3,5,9,12,16,19,21,24,28,31,33,37,40,42,45,49,52,54,58,61,65,68,70,74,
%T A287105 77,79,82,86,89,91,95,98,102,105,107,110,114,117,119,123,126,130,133,
%U A287105 135,139,142,144,147,151,154,156,160,163,167,170,172,175,179,182
%N A287105 Positions of 0 in A287104.
%C A287105 From _Michel Dekking_, Sep 17 2019: (Start)
%C A287105 Let sigma be the defining morphism of A287104: 0->10, 1->12, 2->0.
%C A287105 Let u=01, v=012, w=0121 be the return words of the word 0.
%C A287105 [See Justin & Vuillon (2000) for definition of return word. - _N. J. A. Sloane_, Sep 23 2019]
%C A287105 Then under sigma u, v and w are mapped to
%C A287105     sigma(01) = 1012,  sigma(012) = 10120,  sigma(0121) = 1012012.
%C A287105 Moving the prefix 1 of these three images to the end, the sequence  0 a (i.e., (a(n)) prefixed by the symbol 0), is a fixed point when iterating.
%C A287105 This iteration process induces a morphism 2->4, 3->32, 4->34 on the return words, coded by their lengths.
%C A287105 Coding the symbols according to 2<->2, 4<->0, 3<->1, this leads to the morphism 2->0, 1->12, 0->10 on the alphabet {0,1,2}.
%C A287105 This is simply sigma, which has A287104 as its unique fixed point. So the sequence d of first differences of (a(n)) equals A287104 with the coding above. Noting that the code can be written as x->4-x, this gives the formula below.
%C A287105 (End)
%H A287105 Clark Kimberling, <a href="/A287105/b287105.txt">Table of n, a(n) for n = 1..10000</a>
%H A287105 Jacques Justin and Laurent Vuillon, <a href="http://www.numdam.org/item/ITA_2000__34_5_343_0/">Return words in Sturmian and episturmian words</a>, RAIRO-Theoretical Informatics and Applications 34.5 (2000): 343-356.
%F A287105 a(n) = 4n-1 + Sum_{k=2..n} A287104(k). - _Michel Dekking_, Sep 17 2019
%t A287105 s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 2}, 2 -> 0}] &, {0}, 10] (* A287104 *)
%t A287105 Flatten[Position[s, 0]] (* A287105 *)
%t A287105 Flatten[Position[s, 1]] (* A287106 *)
%t A287105 Flatten[Position[s, 2]] (* A287107 *)
%Y A287105 Cf. A287104, A287106, A287107.
%K A287105 nonn,easy
%O A287105 1,1
%A A287105 _Clark Kimberling_, May 21 2017