A289034 Fixed point of the morphism 0->010, 1->10 starting with 1.

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

Offset

1

Comments

Old name was: 1-limiting word of reversed iterates of the mapping 0->01, 1->0, starting with 0.

The first seven iterates of the mapping 0->01, 1->0, starting with 0, are:

0

01

010

01001

01001010

0100101001001

010010100100101001010

with limit the infinite Fibonacci word A003849.

The corresponding reversed iterates are as follows:

0

10

010

10010

01010010

1001001010010

010100101001001010010

The 1-limiting word is the limit of the n-th iterates for n == 1 mod 2. (The 0-limiting word is A189661.)

There is a simple way to construct this. Let sigma be the time-reversed Fibonacci morphism, i.e., the morphism sigma: 0->10, 1->0. Then sigma^2 is given by 0->010, 1->10, and sigma^2 has two fixed points. The fixed point starting with 1 is the present sequence. - Michel Dekking, Oct 18 2018

Links

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

Example

The first four n-th reversed iterates for n == 1 mod 2 are these:
10
10010
1001001010010
1001001010010010100101001001010010

Mathematica

z = 13;  (*number of iterates*)
s = {0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"0" -> "01", "1" -> "0"}];
r[n_] := StringReverse[w[n]];
TableForm[Table[r[n], {n, 0, 8}]]
rw = ToCharacterCode[r[z]] - 48 (* A289034 *)
p0 = Flatten[Position[rw, 0]];  (* A099267 *)
p1 = Flatten[Position[rw, 1]];  (* A007066 *)

Crossrefs

Cf. A003849, A189661, A099267, A007066, A287523 (binary complement), A188432.

Sequence in context: A068429 A285208 A260444 * A011747 A089013 A188374

Adjacent sequences: A289031 A289032 A289033 * A289035 A289036 A289037

Keyword

nonn,easy

Author

Clark Kimberling, Jul 10 2017

Extensions

Name changed by Michel Dekking, Oct 20 2018

Status

approved