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

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, 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, 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

Offset: 0

Author

Mahdi Saleh, Apr 07 2016

Keywords

• nonn

Comments

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]

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,...

From Michel Dekking, Oct 19 2016: (Start)

The sequence (a(n)) is fixed point of the morphism zeta given by zeta: 0->01, 1->02, 2->epsilon.

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.

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).

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)

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

Links

• Paolo Xausa, Table of n, a(n) for n = 0..13529
• Sam Coates, Metallic mean fractal systems and their tilings, arXiv:2405.04458 [cond-mat.other], 2024. See p. 13.
• F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
• Michel Dekking and Michael Keane, On the conjugacy class of the Fibonacci dynamical system, arXiv preprint arXiv:1608.04487 [Math.DS], 2016.
• Mahdi Saleh and Majid Jahangiri, On linear ternary Intersection sequences and their properties, arXiv:1709.03829 [math.CO], 2017. pages 5-8.

Crossrefs

Cf. A003849, A108103.

Cf. A001622, A019827, A187426, A187798.

Programs

Mathematica

SubstitutionSystem[{0->{0,1}, 1->{0,2}, 2->{}}, {0}, {10}][[1]] (* Paolo Xausa, May 17 2024 *)