A287108 - OEIS

%I #10 May 26 2017 21:31:47

%S 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 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 2,1,1,0,2,1,1,0,1,0,0,2,1,2,1,1,0,1

%N 1-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:  100210212110

%C The 1-limiting word is the limit of the words for which the number of iterations is congruent to 1 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="/A287108/b287108.txt">Table of n, a(n) for n = 1..10000</a>

%e The 1st, 4th, and 7th iterates are

%e 10, 100210212110, 10021021211002121102110100210212110211010021211010021100210212110.

%t s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 1}, 2 -> 0}] &, {0}, 10] (* A287108 *)

%t Flatten[Position[s, 0]] (* A287109 *)

%t Flatten[Position[s, 1]] (* A287110 *)

%t Flatten[Position[s, 2]] (* A287111 *)

%Y Cf. A287109, A287110, A287111.

%K nonn,easy

%O 1,4

%A _Clark Kimberling_, May 21 2017