A143667 - OEIS

A143667

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

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

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.

Wieb Bosma and Henk Don, Constructing Morphisms for Arithmetic Subsequences of Fibonacci, Ch. 6, Logics and Type Systems in Theory and Practice (2024) Lect. Notes Comp. Sci. (LNCS) Vol. 14560, 100-110.

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, The Fibonacci Word Fractal.

Alexis Monnerot-Dumaine, The Fibonacci word fractal [Cached copy, with permission]

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