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A143667 Digits of the infinite Fibonacci word A003849 grouped 2 by 2 and interpreted as a binary value. 4
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Group 2 by 2 the successive letters of the infinite Fibonacci word A003849 then apply: 00->0, 01->1 and 10->2.
Also result of the following iterated morphism: 1->1022, 0->10221, 2->1021, iterated from letter 1. (Monnerot 2008)
Fractal properties studied (proposed for publication)
(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
(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
REFERENCES
M. Lothaire, Combinatorics on words, Cambridge University Press.
LINKS
J.-P. Allouche, M. Mendès France, and G. Skordev, Non-intersectivity of Paperfolding Dragon Curves and of Curves Generated by Automatic Sequences, INTEGERS, Electronic Journal of Combinatorial Number Theory, vol. 18A, Article #A2, 2018. Mentions this sequence.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
Michel Dekking and Mike Keane, Two-block substitutions and morphic words, arXiv:2202.13548 [math.CO], 2022.
A. Monnerot-Dumaine, Fibonacci Fractal
A. Monnerot-Dumaine, The Fibonacci Word Fractal [From Alexis Monnerot-Dumaine (alexis.monnerotdumaine(AT)gmail.com), Aug 31 2009]
J. L. Ramírez and G. N. Rubiano, Properties and Generalizations of the Fibonacci Word Fractal, The Mathematica Journal, Vol. 16 (2014). See "Dense Fibonacci word". - N. J. A. Sloane, Mar 26 2014
FORMULA
a(n) = decimal value of b(2n-1)b(2n), b(n) taken from A003849 (infinite Fibonacci word).
EXAMPLE
a(1) = 1 because the infinite Fibonacci word starts with "01", a(2) = 0 because it continues with "00", and so on.
MATHEMATICA
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 *)
PROG
(Haskell)
a143667 n = a143667_list !! (n-1)
a143667_list = f a003849_list where
f (0:0:ws) = 0 : f ws; f (0:1:ws) = 1 : f ws; f (1:0:ws) = 2 : f ws
-- Reinhard Zumkeller, Jul 29 2014
CROSSREFS
Sequence in context: A282947 A287200 A284387 * A299485 A246785 A084934
KEYWORD
easy,nonn,word
AUTHOR
Alexis Monnerot-Dumaine (alexis.monnerotdumaine(AT)gmail.com), Aug 28 2008
STATUS
approved

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Last modified April 12 07:56 EDT 2024. Contains 371626 sequences. (Running on oeis4.)