A287523 Fixed point starting with 1 of the morphism 0->01, 1->101.

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

Offset

1

Comments

Previous name was: 0-limiting word of reversed iterates of the mapping 0->1, 1->10, starting with 1.

The first eight iterates of the mapping 0->1, 1->10, starting with 1, are these words:

1

10

101

10110

10110101

1011010110110

101101011011010110101

1011010110110101101011011010110110

with limit the infinite Fibonacci word A005614.

The corresponding reversed iterates are as follows:

1

01

101

01101

10101101

0110110101101

101011010110110101101

0110110101101101011010110110101101

The 0-limiting word is the limit of the n-th iterates for n == 0 mod 2. (The 1-limiting word is A189479)

Let sigma be the morphism 0->1, 1->10. Then sigma^2 is given by 0->10, 1->101. The time reversal tau of sigma^2 is given by 0->01, 1->101, and tau^n(1) is equal to the n == 0 mod 2 reversed iterates above. Thus we obtain A189479. - Michel Dekking, Aug 09 2017

The fixed point starting with 0 is A189479. - Michel Dekking, Sep 30 2019

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:
1
101
10101101
101011010110110101101

Mathematica

z = 12;  (*number of iterates*)
s = {0}; w[0] = StringJoin[Map[ToString, s]];
w[n_] := StringReplace[w[n - 1], {"0" -> "1", "1" -> "10"}];
r[n_] := StringReverse[w[n]]; TableForm[Table[r[n], {n, 0, 8}]]
rw = ToCharacterCode[r[z]] - 48; (* A287523 if z even A288243 if z odd *)
p0 = Flatten[Position[rw, 0]];  (* A026356 if z even, A007066 if z odd *)
p1 = Flatten[Position[rw, 1]];  (* A189662 if z even, A099267 if z odd *)

Crossrefs

Cf. A005614, A026356, A189479, A189662.

Sequence in context: A283963 A131670 A188044 * A288932 A155482 A174205

Adjacent sequences: A287520 A287521 A287522 * A287524 A287525 A287526

Keyword

nonn,easy

Author

Clark Kimberling, Jul 10 2017

Extensions

Name changed by Michel Dekking, Sep 30 2019

Status

approved