A105220 Trajectory of 1 under the morphism 1->{1,2,1}, 2->{2,2,2}.

1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 0

Author

Roger L. Bagula, Apr 29 2005

Keywords

• nonn

Comments

Dekking substitution for the Cantor set: characteristic polynomial = x^2 - 5*x + 6 of matrix [2, 0; 1, 3].

This substitution is useful for computing the devil's staircase by bb=aa/. 1->1/3/. 2->0 /. 3->0; ListPlot[FoldList[Plus, 0, bb], PlotRange -> All, PlotJoined -> True, Axes ->False];

The Wikipedia article on L-system Example 3 is "Cantor dust" given by the axiom: A and rules: A -> ABA, B -> BBB. This is isomorphic to the system given in the sequence name. - Michael Somos, Jan 12 2015

Links

• Antti Karttunen, Table of n, a(n) for n = 0..19682
• F. M. Dekking, Recurrent Sets, Advances in Mathematics, vol. 44, no. 1, 1982, page 99, section 4.15
• Wikipedia, L-system Example 3: Cantor dust
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A088917, A316829.

Programs

Mathematica

Flatten[ Nest[ Flatten[ # /. {1 -> {1, 2, 1}, 2 -> {2, 2, 2}} &], {1}, 5]]

PARI

A088917(n) = { while(n, if(n%3==1, return(0), n\=3)); (1); }; \\ Originally from A005823
A105220(n) = (2-A088917(n)); \\ Antti Karttunen, Aug 23 2019

Formula

a(n) = 2 - A088917(n) = 1 + A316829(n). - Antti Karttunen, Aug 24 2019

a(n) = 2 if the ternary expansion of n contains the digit 1, otherwise a(n) = 1. - Joerg Arndt, Aug 24 2019