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A287106 Positions of 1 in A287104.

%I A287106 #14 Oct 03 2019 09:00:48
%S A287106 1,4,6,8,10,13,15,17,20,22,25,27,29,32,34,36,38,41,43,46,48,50,53,55,
%T A287106 57,59,62,64,66,69,71,73,75,78,80,83,85,87,90,92,94,96,99,101,103,106,
%U A287106 108,111,113,115,118,120,122,124,127,129,131,134,136,138,140
%N A287106 Positions of 1 in A287104.
%C A287106 From _Michel Dekking_, Sep 16 2019: (Start)
%C A287106 Let sigma be the defining morphism in A287104: 0->10, 1->12, 2->0.
%C A287106 Let u := 10, v := 12, w: = 120 be the return words of the word 1. [See Justin & Vuillon (2000) for definition of return word. - _N. J. A. Sloane_, Sep 23 2019]
%C A287106 Then
%C A287106     sigma(u) = vu, sigma(v) = w, sigma(w) = wu.
%C A287106 If we code w<->0, u<->1, v<->2, then this morphism turns into the morphism
%C A287106     0 -> 01, 1 -> 21, 2 -> 0.
%C A287106 This is exactly the morphism which has A287072 as unique fixed point.
%C A287106 Since u and v have length 2 and w has length 3, this implies that the sequence d of first differences of (a(n)) equals A287072 with the projection 0 -> 3, 1 -> 2, 2 -> 2. This gives the formula below.
%C A287106 (End)
%H A287106 Clark Kimberling, <a href="/A287106/b287106.txt">Table of n, a(n) for n = 1..10000</a>
%H A287106 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 A287106 a(n) = 1 + Sum_{k=1..n-1} d(k), where d(k) = 3 if A287072(k)=0, and d(k) = 2 otherwise, for k = 1,...,n. - _Michel Dekking_, Sep 16 2019
%t A287106 s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 2}, 2 -> 0}] &, {0}, 10] (* A287104 *)
%t A287106 Flatten[Position[s, 0]] (* A287105 *)
%t A287106 Flatten[Position[s, 1]] (* A287106 *)
%t A287106 Flatten[Position[s, 2]] (* A287107 *)
%Y A287106 Cf. A287104, A287105, A287107. Closely related to A287072.
%K A287106 nonn,easy
%O A287106 1,2
%A A287106 _Clark Kimberling_, May 21 2017