cp's OEIS Frontend

A143667 Digits of the infinite Fibonacci word A003849 grouped 2 by 2 and interpreted as a binary value.

%I A143667 #52 Dec 26 2024 10:37:50
%S A143667 1,0,2,2,1,0,2,2,1,1,0,2,1,1,0,2,1,1,0,2,2,1,0,2,2,1,1,0,2,1,1,0,2,1,
%T A143667 1,0,2,2,1,0,2,2,1,0,2,2,1,1,0,2,1,1,0,2,2,1,0,2,2,1,0,2,2,1,1,0,2,1,
%U A143667 1,0,2,2,1,0,2,2,1,0,2,2,1,1,0,2,1,1,0,2,1,1,0,2,2,1,0,2,2,1,1,0
%N A143667 Digits of the infinite Fibonacci word A003849 grouped 2 by 2 and interpreted as a binary value.
%C A143667 Group 2 by 2 the successive letters of the infinite Fibonacci word A003849 then apply: 00->0, 01->1 and 10->2.
%C A143667 Also result of the following iterated morphism: 1->1022, 0->10221, 2->1021, iterated from letter 1. (Monnerot 2008)
%C A143667 Fractal properties studied (proposed for publication)
%C A143667 (a(n)) is essentially the same sequence as A123564. Simply change the alphabet to {1,2,3}, and permute the letters. The Standard Form of (a(n)) written as a word on the alphabet {a,b,c} is abccabccaabc... . Other forms for this standard form are 1,2,3,3,1,2,3,3,1,1,2,3,.... and 123312331123... - __Michel Dekking_, Oct 07 2017
%C A143667 (a(n)) is the fixed point of the 2-block map (called 2-block Fibonacci to the power 3) 00->0100101001, 01->01001010, 10->01001001, followed by the coding above. - _Michel Dekking_, Sep 26 2017
%D A143667 M. Lothaire, Combinatorics on words, Cambridge University Press.
%H A143667 Reinhard Zumkeller, <a href="/A143667/b143667.txt">Table of n, a(n) for n = 1..1000</a>
%H A143667 J.-P. Allouche, M. Mendès France, and G. Skordev, <a href="http://math.colgate.edu/~integers/sjs2/sjs2.Abstract.html">Non-intersectivity of Paperfolding Dragon Curves and of Curves Generated by Automatic Sequences</a>, INTEGERS, Electronic Journal of Combinatorial Number Theory, vol. 18A, Article #A2, 2018. Mentions this sequence.
%H A143667 Wieb Bosma and Henk Don, <a href="https://doi.org/10.1007/978-3-031-61716-4_6">Constructing Morphisms for Arithmetic Subsequences of Fibonacci</a>, Ch. 6, Logics and Type Systems in Theory and Practice (2024) Lect. Notes Comp. Sci. (LNCS) Vol. 14560, 100-110.
%H A143667 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 A143667 Michel Dekking and Mike Keane, <a href="https://arxiv.org/abs/2202.13548">Two-block substitutions and morphic words</a>, arXiv:2202.13548 [math.CO], 2022.
%H A143667 A. Monnerot-Dumaine, <a href="https://hal.science/hal-00367972">The Fibonacci Word Fractal</a>.
%H A143667 Alexis Monnerot-Dumaine, <a href="/A171587/a171587.pdf">The Fibonacci word fractal</a> [Cached copy, with permission]
%H A143667 J. L. Ramírez and G. N. Rubiano, <a href="http://www.mathematica-journal.com/2014/02/properties-and-generalizations-of-the-fibonacci-word-fractal/">Properties and Generalizations of the Fibonacci Word Fractal</a>, The Mathematica Journal, Vol. 16 (2014). See "Dense Fibonacci word". - _N. J. A. Sloane_, Mar 26 2014
%F A143667 a(n) = decimal value of b(2n-1)b(2n), b(n) taken from A003849 (infinite Fibonacci word).
%e A143667 a(1) = 1 because the infinite Fibonacci word starts with "01", a(2) = 0 because it continues with "00", and so on.
%t A143667 Table[3 - (Floor[#1 #2] - 2 Floor[#1 (#2 - 1)] + Floor[#1 (#2 + 1)]) & @@ {1/GoldenRatio, 2 n}, {n, 100}] (* _Michael De Vlieger_, Oct 06 2017 *)
%o A143667 (Haskell)
%o A143667 a143667 n = a143667_list !! (n-1)
%o A143667 a143667_list = f a003849_list where
%o A143667    f (0:0:ws) = 0 : f ws; f (0:1:ws) = 1 : f ws; f (1:0:ws) = 2 : f ws
%o A143667 -- _Reinhard Zumkeller_, Jul 29 2014
%Y A143667 Cf. A003849, A123564.
%K A143667 easy,nonn,word
%O A143667 1,3
%A A143667 _Alexis Monnerot-Dumaine_, Aug 28 2008