cp's OEIS Frontend

A287104 Start with 0 and repeatedly substitute 0->10, 1->12, 2->0.

%I A287104 #19 Jul 18 2025 10:13:19
%S A287104 1,2,0,1,0,1,2,1,0,1,2,0,1,2,1,0,1,2,0,1,0,1,2,0,1,2,1,0,1,2,0,1,0,1,
%T A287104 2,1,0,1,2,0,1,0,1,2,0,1,2,1,0,1,2,0,1,0,1,2,1,0,1,2,0,1,2,1,0,1,2,0,
%U A287104 1,0,1,2,1,0,1,2,0,1,0,1,2,0,1,2,1,0
%N A287104 Start with 0 and repeatedly substitute 0->10, 1->12, 2->0.
%C A287104 The fixed point of the morphism 0->10, 1->12, 2->0.  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.  It appears that 1/U + 1/V + 1/W = 1, where
%C A287104 U = 3.079595623491438786010417...,
%C A287104 V = 2.324717957244746025960908...,
%C A287104 W = U + 1 = 4.079595623491438786010417....
%C A287104 From _Michel Dekking_, Sep 15 2019: (Start)
%C A287104 The incidence matrix of the morphism sigma: 0->10, 1->12, 2->0  has characteristic polynomial  chi(u) = u^3-2u^2+u-1. The real root of chi is lambda :=  Q/6 + 2/3*1/Q + 2/3, where
%C A287104     Q = ( 100 + 12*sqrt(69) )^1/3.
%C A287104 An eigenvector of lambda is  (1, lambda^2-lambda, lambda-1).
%C A287104 The Perron-Frobenius Theorem gives that the asymptotic frequencies f0, f1 and f2 of the letters  0, 1, and 2 are
%C A287104     f0 = 1/lambda^2,
%C A287104     f1 = (lambda^2 - lambda +1)/lambda^3,
%C A287104     f2 = (lambda - 1)/lambda^2.
%C A287104 Algebraic expressions for the constants U,V and W are then given by
%C A287104     U = 1/f0,  V = 1/f1, W = 1/f2.
%C A287104 In particular, this shows that W = U + 1.
%C A287104 (End)
%C A287104 Conjecture: if n >=2, then u(n) - u(n-1) is in {2,3,4}, v(n) - v(n-1) is in {2,3}, and w(n) - w(n-1) is in {3,4,5}.
%C A287104 See A287105, A287106, and A287107 for proofs of these conjectures, with explicit expressions for u, v, and w. - _Michel Dekking_, Sep 15 2019
%H A287104 Clark Kimberling, <a href="/A287104/b287104.txt">Table of n, a(n) for n = 1..10000</a>
%H A287104 James D. Currie, <a href="https://arxiv.org/abs/2507.09387">Words with factor somplexity 2n+1 and minimal critical exponent</a>, arXiv:2507.09387 [math.CO], 2025. See p. 2.
%H A287104 <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%t A287104 s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 2}, 2 -> 0}] &, {0}, 10] (* A287104 *)
%t A287104 Flatten[Position[s, 0]] (* A287105 *)
%t A287104 Flatten[Position[s, 1]] (* A287106 *)
%t A287104 Flatten[Position[s, 2]] (* A287107 *)
%Y A287104 Cf. A287105, A287106, A287107.
%K A287104 nonn,easy
%O A287104 1,2
%A A287104 _Clark Kimberling_, May 21 2017