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A287108 1-limiting word of the morphism 0->10, 1->21, 2->0.

%I A287108 #10 May 26 2017 21:31:47
%S A287108 1,0,0,2,1,0,2,1,2,1,1,0,0,2,1,2,1,1,0,2,1,1,0,1,0,0,2,1,0,2,1,2,1,1,
%T A287108 0,2,1,1,0,1,0,0,2,1,2,1,1,0,1,0,0,2,1,1,0,0,2,1,0,2,1,2,1,1,0,0,2,1,
%U A287108 2,1,1,0,2,1,1,0,1,0,0,2,1,2,1,1,0,1
%N A287108 1-limiting word of the morphism 0->10, 1->21, 2->0.
%C A287108 Starting with 0, the first 4 iterations of the morphism yield words shown here:
%C A287108 1st:  10
%C A287108 2nd:  2110
%C A287108 3rd:  0212110
%C A287108 4th:  100210212110
%C A287108 The 1-limiting word is the limit of the words for which the number of iterations is congruent to 1 mod 3.
%C A287108 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 A287108 U = 3.079595623491438786010417...,
%C A287108 V = 2.324717957244746025960908...,
%C A287108 W = U + 1 = 4.079595623491438786010417....
%C A287108 If n >=2, then u(n) - u(n-1) is in {1,2,3,4,6}, v(n) - v(n-1) is in {1,2,3,4}, and w(n) - w(n-1) is in {2,3,4,5,7}.
%H A287108 Clark Kimberling, <a href="/A287108/b287108.txt">Table of n, a(n) for n = 1..10000</a>
%e A287108 The 1st, 4th, and 7th iterates are
%e A287108 10, 100210212110, 10021021211002121102110100210212110211010021211010021100210212110.
%t A287108 s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 10] (* A287108 *)
%t A287108 Flatten[Position[s, 0]] (* A287109 *)
%t A287108 Flatten[Position[s, 1]] (* A287110 *)
%t A287108 Flatten[Position[s, 2]] (* A287111 *)
%Y A287108 Cf. A287109, A287110, A287111.
%K A287108 nonn,easy
%O A287108 1,4
%A A287108 _Clark Kimberling_, May 21 2017