A285076 1-limiting word of the morphism 0->10, 1-> 010.

1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1

Offset

1

Comments

The morphism 0->10, 1->010 has two limiting words. If the number of iterations is even, the 0-word evolves from 0 -> 10 -> 01010 -> 100101001010 -> 01010100101001010100101001010; if the number of iterations is odd, the 1-word evolves from 0 -> 10 -> 01010 -> 100101001010, as in A285076.

From Michel Dekking, May 27 2017: (Start)

(a(n-1)) is the Sturmian sequence

s(r,1-r) = (floor((n+1)r+1-r)-floor(nr+1-r))) with r = sqrt(2)-1.

Moreover, a = 10c, where c = A159684 is the Sturmian sequence

s(r,r) = (floor((n+2)r)-floor((n+1)r))),

fixed point of the morphism gamma: 0 -> 01, 1 -> 010.

The morphism gamma is the time-reversal of the morphism psi: 0 -> 10, 1 -> 010 generating (a(n)).

This is a general property: see reference and link. It is also general that the square of psi has a second fixed point b = 01c, where b = A285073. (End)

Links

Clark Kimberling, Table of n, a(n) for n = 1..10000

J. Berstel et P. Séébold, A remark on Sturmian words, RAIRO - Theoretical Informatics and Applications 28 (1994), 255-263.

Michel Dekking, Substitution invariant Sturmian words and binary trees, arXiv:1705.08607 [math.CO], 2017.

Michel Dekking, Substitution invariant Sturmian words and binary trees, Integers, Electronic Journal of Combinatorial Number Theory 18A (2018), #A17.

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {0, 1, 0}}] &, {0}, 13]; (* A285076 *)
Flatten[Position[s, 0]];  (* A285077 *)
Flatten[Position[s, 1]];  (* A285078 *)

Crossrefs

Cf. A285074, A285077, A285078.

Sequence in context: A213728 A105367 A232990 * A338353 A267598 A194681

Adjacent sequences: A285073 A285074 A285075 * A285077 A285078 A285079

Keyword

nonn,easy

Author

Clark Kimberling, Apr 19 2017

Status

approved