A189664 Fixed point of the morphism 0->010, 1->001.

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

Offset

1

Comments

a(n) is the parity of the number of ternary 1-digits below the lowest 0-digit in the ternary expansion of n-1. All fixed-width morphisms have a similar digital interpretation. - Kevin Ryde, Apr 26 2017

Links

Joerg Arndt, Table of n, a(n) for n = 1..2187

Index entries for sequences that are fixed points of mappings

Formula

a(3k-2)=0, a(3k-1)=1-a(k), a(3k)=a(k) for k>=1, a(0)=0.

2*a(n) = A284794(n) for n >= 1. - Clark Kimberling, Apr 14 2017

This formula cannot be correct, it belongs to A189665. - Michel Dekking, Oct 15 2019

a(n) = A284777(n) - 2n. - Clark Kimberling, Apr 15 2017

Example

Iterating the morphism starting with 0:
0:   (#=1)
  0
1:   (#=3)
  010
2:   (#=9)
  010001010
3:   (#=27)
  010001010010010001010001010
4:   (#=81)
  010001010010010001010001010010001010010001010010010001010001010010010001010001010
etc.

Mathematica

t = Nest[Flatten[# /. {0->{0, 1, 0}, 1->{0, 0, 1}}] &, {0}, 5] (* A189664 *)
f[n_] := t[[n]]
Flatten[Position[t, 0]] (* A189665 *)
Flatten[Position[t, 1]] (* A189666 *)
s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;
Table[s[n], {n, 1, 120}] (* A189667 *)
Nest[Flatten[# /. a_Integer -> {0, Abs[a - 1], a}] &, {0}, 5] (* Robert G. Wilson v, Jul 16 2012 *)

Prog

(PARI) a(n) = n--; my(ret=0); while(n%3, if(n%3==1, ret=!ret); n\=3); ret;  /* Kevin Ryde, Jul 23 2019 */

Crossrefs

Cf. A189628, A189665, A189666, A189667.

Sequence in context: A354354 A117943 A285969 * A260446 A096268 A219097

Adjacent sequences: A189661 A189662 A189663 * A189665 A189666 A189667

Keyword

nonn

Author

Clark Kimberling, Apr 25 2011

Status

approved