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%I #18 Sep 21 2022 09:48:37
%S 0,1,2,1,0,2,1,0,1,0,2,1,0,1,2,1,0,1,0,2,1,0,1,2,1,0,2,1,0,1,2,1,0,1,
%T 0,2,1,0,1,2,1,0,2,1,0,1,0,2,1,0,1,2,1,0,2,1,0,1,2,1,0,1,0,2,1,0,1,2,
%U 1,0,2,1,0,1,0,2,1,0,1,2,1,0,1,0,2,1
%N Start with 0 and repeatedly substitute 0->01, 1->21, 2->0.
%C A fixed point of the morphism 0->01, 1->21, 2->0. Let u be the sequence of positions of 0, and likewise, v for 1 and w for 2. Let U, V, W be the limits of u(n)/n, v(n)/n, w(n)/n, respectively. Then 1/U + 1/V + 1/W = 1, where
%C U = 3.079595623491438786010417...,
%C V = 2.324717957244746025960908...,
%C W = U + 1 = 4.079595623491438786010417....
%C Since the morphism 0->01, 1->21, 2->0 is the time reversal of the morphism 0->10, 1->12 2->0, which has fixed point A287104, in particular the incidence matrices of these two morphisms are equal. Thus the algebraic expressions found for U, V and W in A287104 do also apply to the U, V and W above. - _Michel Dekking_, Sep 15 2019
%C If n >=2, then u(n) - u(n-1) is in {2,3,4}, v(n) - v(n-1) is in {2,3}, and w(n) - w(n-1) is in {3,4,5}.
%H Clark Kimberling, <a href="/A287072/b287072.txt">Table of n, a(n) for n = 1..10000</a>
%H James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, <a href="https://arxiv.org/abs/2206.01776">Properties of a Ternary Infinite Word</a>, arXiv:2206.01776 [cs.DM], 2022.
%H James Currie, Pascal Ochem, Narad Rampersad, and Jeffrey Shallit, <a href="https://arxiv.org/abs/2209.09598">Complement Avoidance in Binary Words</a>, arXiv:2209.09598 [math.CO], 2022.
%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 10] (* A287072 *)
%t Flatten[Position[s, 0]] (* A287073 *)
%t Flatten[Position[s, 1]] (* A287074 *)
%t Flatten[Position[s, 2]] (* A287075 *)
%Y Cf. A057985, A287073, A287074, A287075, A287104.
%K nonn,easy
%O 1,3
%A _Clark Kimberling_, May 21 2017
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