cp's OEIS Frontend

A267860 An infinite ternary 3-Fibonacci sequence (replace each 00 factor of the Fibonacci word with 020).

%I A267860 #52 May 28 2024 06:47:38
%S A267860 0,1,0,2,0,1,0,1,0,2,0,1,0,2,0,1,0,1,0,2,0,1,0,1,0,2,0,1,0,2,0,1,0,1,
%T A267860 0,2,0,1,0,2,0,1,0,1,0,2,0,1,0,1,0,2,0,1,0,2,0,1,0,1,0,2,0,1,0,1,0,2,
%U A267860 0,1,0,2,0,1,0,1,0,2,0,1,0,2,0,1,0,1,0,2,0,1,0,1,0,2,0,1,0,2,0,1,0,1,0,2,0,1,0,2,0,1,0,1,0,2,0,1,0,1,0,2,0,1
%N A267860 An infinite ternary 3-Fibonacci sequence (replace each 00 factor of the Fibonacci word with 020).
%C A267860 A word constructed by replacing each 00 factor of the Fibonacci word (A003849) with 020. The obtained ternary sequence is a word with Sturmian erasures (by removing each word,the obtained binary sequence is Sturmian)[1]. By removing each of 0's or 2's, the set of replacements on the Fibonacci word, is equal to the morphisms of deriving the Fibonacci word [2]. So the obtained binary word by removing each of 0's,1's or 2's is the Fibonacci word. Since the slope of the sequential projection (sending for example one letter to 1 and all the others to 0) is 1, the factor complexity of this ternary word for each integer n>0, is n+2.[3]
%C A267860 The binary sequence obtained by removing all 0's from the 3-Fibonacci word: 1,2,1,1,2,1,2,1,1,2,1,2,1,1,2,1,...
%C A267860 From _Michel Dekking_, Oct 19 2016: (Start)
%C A267860 The sequence (a(n)) is fixed point of the morphism zeta given by zeta: 0->01, 1->02, 2->epsilon.
%C A267860 Here epsilon is the empty word. To see this, code the 0’s in the Fibonacci sequence followed by 0 by 5, and the 0’s followed by 1 by 6. Then add 2 after 5. This gives the morphism 1->52, 5->61, 6->61, 2->epsilon. Then injectivize, i.e., map 5 and 6 to 0.
%C A267860 The sequence (a(n)) is related to A108103. Let theta be the standard form of zeta: theta(1)=12, theta(2)=13, theta(3)=epsilon. Let psi be the morphism generating the version of A108103 with 2 and 3 interchanged, psi: 1->2, 2->131, 3->1. Then the unique fixed point of theta is different from the fixed points of psi, but theta and psi generate the same language, i.e., arbitrarily long words occurring in the fixed point of theta occur in the fixed points of psi. This is a nontrivial exercise (prove that 2 theta^{2n}(1) = psi^{2n}(2) 13 for all n>0).
%C A267860 The sequence (a(n)) is not related to A270788, which might be called the ternary Fibonacci sequence. The dynamical system generated by (a(n)) has an eigenvalue -1, whereas the system generated by A270788 is isomorphic to the Fibonacci dynamical system. (End)
%C A267860 The asymptotic density of the occurrences of 0, 1, and 2 is 1/2, 1/(2*phi) = A019827, and 1/(2*phi^2) = A187426 / 10, respectively, where phi is the golden ratio (A001622). The asymptotic mean of this sequence is (3-phi)/2 (A187798). - _Amiram Eldar_, May 28 2024
%H A267860 Paolo Xausa, <a href="/A267860/b267860.txt">Table of n, a(n) for n = 0..13529</a>
%H A267860 Sam Coates, <a href="https://arxiv.org/abs/2405.04458">Metallic mean fractal systems and their tilings</a>, arXiv:2405.04458 [cond-mat.other], 2024. See p. 13.
%H A267860 F. Michel Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Dekking/dekk4.html">Morphisms, Symbolic Sequences, and Their Standard Forms</a>, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
%H A267860 Michel Dekking and Michael Keane, <a href="http://arxiv.org/abs/1608.04487">On the conjugacy class of the Fibonacci dynamical system</a>, arXiv preprint arXiv:1608.04487 [Math.DS], 2016.
%H A267860 Mahdi Saleh and Majid Jahangiri, <a href="https://arxiv.org/abs/1709.03829">On linear ternary Intersection sequences and their properties</a>, arXiv:1709.03829 [math.CO], 2017. pages 5-8.
%t A267860 SubstitutionSystem[{0->{0,1}, 1->{0,2}, 2->{}}, {0}, {10}][[1]] (* _Paolo Xausa_, May 17 2024 *)
%Y A267860 Cf. A003849, A108103.
%Y A267860 Cf. A001622, A019827, A187426, A187798.
%K A267860 nonn
%O A267860 0,4
%A A267860 _Mahdi Saleh_, Apr 07 2016