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%I #16 Oct 14 2019 00:31:22
%S 1,1,0,1,1,0,0,2,0,2,1,1,0,1,1,0,0,2,0,2,0,2,0,2,1,1,0,2,0,2,1,1,1,1,
%T 0,1,1,0,0,2,0,2,1,1,0,1,1,0,0,2,0,2,0,2,0,2,1,1,0,2,0,2,1,1,0,2,0,2,
%U 1,1,0,2,0,2,1,1,1,1,0,1,1,0,0,2,0,2
%N 1-limiting word of the morphism 0->11, 1->02, 2->0.
%C Starting with 0, the first 5 iterations of the morphism yield words shown here:
%C 1st: 11
%C 2nd: 0202
%C 3rd: 110110
%C 4th: 020211020211
%C 5th: 11011002021101100202
%C The 1-limiting word is the limit of the words for which the number of iterations is odd.
%C 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 = 2.7692923542386314152404094643350334926...,
%C V = 2.4498438945029551040577327454145475624...,
%C W = 4.3344900716222708116779374775820643087...
%C If n >=2, then u(n) - u(n-1) is in {1,2,3,4,6}, v(n) - v(n-1) is in {1,2,5,6,10}, and w(n) - w(n-1) is in {2,4,8,10,16}.
%C From _Michel Dekking_, Oct 09 2019: (Start)
%C Let u, v and w be the positions of 0, 1 and 2 in this sequence.
%C The incidence matrix of the defining morphism: 0->10, 1->12, 2->0 has characteristic polynomial chi(u) = u^3 - 2u - 2.
%C Let Q = [27+ 3*sqrt(57)]^(1/3). Then the real root of the characteristic polynomial chi is lambda := Q/3 + 2/Q.
%C An eigenvector of lambda is (1, lambda^2-2, -lambda^2+lambda+2).
%C The Perron-Frobenius Theorem then gives that the asymptotic frequencies f0, f1 and f2 of the letters 0, 1, and 2 are f0 = 1/(1+lambda), f1 = (lambda^2-2)/(1+lambda), and f2 = (-lambda^2+lambda+2)/(1+lambda).
%C Algebraic expressions for the three constants U, V and W are then given by U = 1/f0, V = 1/f1, W = 1/f2.
%C (End)
%H Clark Kimberling, <a href="/A287267/b287267.txt">Table of n, a(n) for n = 1..10000</a>
%e 3rd iterate: 110110;
%e 5th iterate: 11011002021101100202.
%t s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {0, 2}, 2 -> 0}] &, {0}, 11] (* A287267 *)
%t Flatten[Position[s, 0]] (* A287268 *)
%t Flatten[Position[s, 1]] (* A287269 *)
%t Flatten[Position[s, 2]] (* A287270 *)
%Y Cf. A287263 (0-limiting word), A287268, A287269, A287270.
%K nonn,easy
%O 1,8
%A _Clark Kimberling_, May 24 2017
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