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%I #5 May 22 2017 12:23:06
%S 2,1,1,0,1,0,0,2,1,1,0,0,2,1,0,2,1,2,1,1,0,1,0,0,2,1,0,2,1,2,1,1,0,0,
%T 2,1,2,1,1,0,2,1,1,0,1,0,0,2,1,1,0,0,2,1,0,2,1,2,1,1,0,0,2,1,2,1,1,0,
%U 2,1,1,0,1,0,0,2,1,0,2,1,2,1,1,0,2,1
%N 2-limiting word of the morphism 0->10, 1->21, 2->0.
%C Starting with 0, the first 4 iterations of the morphism yield words shown here:
%C 1st: 10
%C 2nd: 2110
%C 3rd: 0212110
%C 4th: 1002010212110
%C The 2-limiting word is the limit of the words for which the number of iterations is congruent to 2 mod 3.
%C 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 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 Clark Kimberling, <a href="/A287156/b287156.txt">Table of n, a(n) for n = 1..10000</a>
%e The 2nd, 5th, and 8th iterates are 2110, 211010021100210212110, 211010021100210212110100210212110021211021101002110021021211002121102110100210212110211010021211010021100210212110.
%t s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 11] (* A287156 *)
%t Flatten[Position[s, 0]] (* A287157 *)
%t Flatten[Position[s, 1]] (* A287158 *)
%t Flatten[Position[s, 2]] (* A287159 *)
%Y Cf. A287108, A287157, A287158, A287159, A297160.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, May 22 2017
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