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A287112 1-limiting word of the morphism 0->10, 1->20, 2->0.

%I A287112 #12 Jul 21 2021 07:16:45
%S A287112 1,0,2,0,1,0,0,1,0,2,0,1,0,2,0,1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,
%T A287112 0,1,0,0,1,0,2,0,1,0,1,0,2,0,1,0,0,1,0,2,0,1,0,2,0,1,0,0,1,0,2,0,1,0,
%U A287112 1,0,2,0,1,0,0,1,0,2,0,1,0,2,0,1,0,0
%N A287112 1-limiting word of the morphism 0->10, 1->20, 2->0.
%C A287112 Starting with 0, the first 4 iterations of the morphism yield words shown here:
%C A287112 1st:  10
%C A287112 2nd:  2010
%C A287112 3rd:  0102010
%C A287112 4th:  1020100102010
%C A287112 The 1-limiting word is the limit of the words for which the number of iterations is congruent to 1 mod 3.
%C A287112 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 A287112 U = 1.8392867552141611325518525646532866...,
%C A287112 V = U^2 = 3.3829757679062374941227085364...,
%C A287112 W = U^3 = 6.2222625231203986266745611011....
%C A287112 If n >=2, then u(n) - u(n-1) is in {1,2}, v(n) - v(n-1) is in {2,3,4}, and w(n) - w(n-1) is in {4,6,7}.
%C A287112 From _Michel Dekking_, Mar 29 2019: (Start)
%C A287112 This sequence is one of the three fixed points of the morphism alpha^3, where alpha is the defining morphism
%C A287112       0->10, 1->20, 2->0.
%C A287112 The other two fixed points are A286998 and A287174.
%C A287112 We have alpha = rho(tau), where tau is the tribonacci morphism in A080843
%C A287112       0->01, 1->02, 2->0,
%C A287112 and rho is the rotation operator.
%C A287112 An eigenvector computation of the incidence matrix of the morphism gives that 0,1, and 2 have frequencies 1/t, 1/t^2 and 1/t^3, where t is the tribonacci constant A058265.
%C A287112 Apparently (u(n)) := A287113. Thus U, the limit of u(n)/n, equals 1/(1/t), the tribonacci constant t. Also, V = A276800, and W = A276801.
%C A287112 (End)
%H A287112 Clark Kimberling, <a href="/A287112/b287112.txt">Table of n, a(n) for n = 1..10000</a>
%e A287112 1st iterate: 10
%e A287112 4th iterate: 1020100102010
%e A287112 7th iterate:  102010010201020100102010102010010201001020101020100102010201001020101020100102010.
%t A287112 s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {2, 0}, 2 -> 0}] &, {0}, 10]   (* A287112 *)
%t A287112 Flatten[Position[s, 0]] (* A287113 *)
%t A287112 Flatten[Position[s, 1]] (* A287114 *)
%t A287112 Flatten[Position[s, 2]] (* A287115 *)
%Y A287112 Cf. A287113, A287114, A287115, A286998, A287174.
%K A287112 nonn,easy
%O A287112 1,3
%A A287112 _Clark Kimberling_, May 22 2017