A189668 Fixed point of the morphism 0->010, 1->100.

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

Offset

1

Links

Table of n, a(n) for n=1..122.

Mazen Khodier, New Methods for Analyzing the Properties of Automatic Sequences, Master's Thesis, Univ. Waterloo (Canada 2026). See p. 27, Table 5.1.

Index entries for sequences that are fixed points of mappings

Formula

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

Conjecture: a(n) = 2*n - 1 - A285347(n).

This conjecture is correct, and proved in A285347. - Michel Dekking, Sep 07 2022

Example

0->010->010100010->

Mathematica

t = Nest[Flatten[# /. {0->{0, 1, 0}, 1->{1, 0, 0}}] &, {0}, 5] (*A189668*)
f[n_] := t[[n]]
Flatten[Position[t, 0]] (*A189669*)
Flatten[Position[t, 1]] (*A189670*)
s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;
Table[s[n], {n, 1, 120}] (*A189671*)

Crossrefs

Cf. A189628, A189669, A189670, A189671, A285347.

Sequence in context: A353674 A288512 A157687 * A353525 A353529 A286907

Adjacent sequences: A189665 A189666 A189667 * A189669 A189670 A189671

Keyword

nonn

Author

Clark Kimberling, Apr 25 2011

Status

approved