A189628 Fixed point of the morphism 0->001, 1->010.

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, 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, 0

Offset

1

Comments

A189628 is one of many 01-sequences fixed by morphisms. An extension of the list begun at A189576 is continued here with sequences of type 3,3.

Each row shows a morphism, followed by four sequences:

(1) the fixed sequence a [starting from a(0)=0],

(2) positions of 0 in a,

(3) positions of 1 in a,

(4) partial sums of a.

Some lower-numbered entries are conjectural.

0->001, 1->010..A189628..A189629..A189630..A189631

0->001, 1->011..A116178..A189636..A189637..A189638

0->001, 1->100..A189632..A189633..A189634..A189635

0->001, 1->101..A189640..A026138..A026323..A189641

0->001, 1->110..A064990..A189658..A189659..A189660

0->010, 1->001..A189664..A189665..A189666..A189667

0->010, 1->011..A080846..A026225..A026179..A189672

0->010, 1->100..A189668..A189669..A189670..A189671

0->010, 1->101..A000035..A005408..A005843..A004526

0->010, 1->110..A189673..A026227..A026138..A189674

0->011, 1->001..A189706..A189707..A189708..A189709

0->011, 1->010..A156595..A189715..A189716..A189717

0->011, 1->100..A189718..A189719..A189720..A189721

0->011, 1->101..A189723..A189724..A189725..A189726

0->011, 1->110..A189727..A189728..A189729..A189730

Each of the 15 sequences in column 3 (i.e., A189628 to A189727) is generated by a 3-part recurrence, as in the Formula section. Two other sequences generated by such a recurrence are A189816 and A189820.

References

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.

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(3k-2)=0, a(3k-1)=a(k), a(3k)=1-a(k) for k>=1. - corrected by Michel Dekking, Sep 09 2022

Example

0->001->001001010->->

Mathematica

t = Nest[Flatten[# /. {0->{0, 0, 1}, 1->{0, 1, 0}}] &, {0}, 5] (*A189628*)
f[n_] := t[[n]]
Flatten[Position[t, 0]] (*A189629*)
Flatten[Position[t, 1]] (*A189630*)
s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;
Table[s[n], {n, 1, 120}] (*A189631*)

Crossrefs

Cf. A189629, A189630, A189631, A189576.

Sequence in context: A143518 A122414 A288216 * A289239 A188432 A282317

Adjacent sequences: A189625 A189626 A189627 * A189629 A189630 A189631

Keyword

nonn

Author

Clark Kimberling, Apr 24 2011

Status

approved