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.

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