A287458 -id:A287458

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A287385

Start with 0 and repeatedly substitute 0->012, 1->210, 2->021.

+10
19

0, 1, 2, 2, 1, 0, 0, 2, 1, 0, 2, 1, 2, 1, 0, 0, 1, 2, 0, 1, 2, 0, 2, 1, 2, 1, 0, 0, 1, 2, 0, 2, 1, 2, 1, 0, 0, 2, 1, 2, 1, 0, 0, 1, 2, 0, 1, 2, 2, 1, 0, 0, 2, 1, 0, 1, 2, 2, 1, 0, 0, 2, 1, 0, 1, 2, 0, 2, 1, 2, 1, 0, 0, 2, 1, 2, 1, 0, 0, 1, 2, 0, 1, 2, 2, 1

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

This is the fixed point of the morphism 0->012, 1->210, 2->021 starting with 0. Let u be the (nonperiodic) sequence of positions of 0, and likewise, v for 1 and w for 2; then u(n)/n -> 3, v(n)/n -> 3, w(n)/n -> 3.

In the following guide to related sequences, column 1 indexes fixed points on {1,2,3}, and columns 2,3,4 match the position sequences of 0, 1, 2. Those sequences therefore comprise a 3-way splitting of the positive integers.

Fixed point and morphism Position sequences

A287385: 0->012, 1->210, 2->021 A287386 A287387 A287388

A287397: 0->012, 1->210, 2->102 A287398 A287399 A287400

A287401: 0->012, 1->210, 2->120 A189728 A287403 A287404

A287407: 0->012, 1->210, 2->201 A287408 A287409 A287410

A287411: 0->012, 1->120, 2->021 A287412 A287413 A287414

A287418: 0->012, 1->120, 2->102 A287419 A287420 A287421

A053838: 0->012, 1->120, 2->201 A287435 A287436 A287437

A287438: 0->012, 1->120, 2->210 A189728 A189670 A287441

A287443: 0->012, 1->201, 2->021 A287444 A287445 A287446

A287447: 0->012, 1->201, 2->102 A189724 A287449 A287450

A287451: 0->012, 1->201, 2->120 A287452 A287453 A287454

A287455: 0->012, 1->201, 2->210 A287456 A189666 A287458

A287516: 0->012, 1->102, 2->021 A287517 A287518 A189630

A287520: 0->012, 1->102, 2->120 A287521 A287522 A189630

A287524: 0->012, 1->102, 2->201 A189724 A287526 A287527

A287528: 0->012, 1->102, 2->210 A287529 A189670 A189634

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000

Index entries for sequences that are fixed points of mappings

EXAMPLE

First three iterations of the morphism: 012, 012210021, 012210021021210012012021210.

MATHEMATICA

s = Nest[Flatten[# /. {0->{0, 1, 2}, 1->{2, 1, 0}, 2->{0, 2, 1}}] &, {0}, 9]; (*A287385*)

Flatten[Position[s, 0]]; (*A287386*)

Flatten[Position[s, 1]]; (*A287387*)

Flatten[Position[s, 2]]; (*A287388*)

CROSSREFS

Cf. A287386, A287387, A287388.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 25 2017

EXTENSIONS

Two entries in overview corrected by Georg Fischer, Sep 20 2021

STATUS

approved

A287455

Start with 0 and repeatedly substitute 0->012, 1->201, 2->210.

+10
4

0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 1, 0, 0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 0, 1, 0, 1, 2, 2, 1, 0, 2, 0, 1, 0, 1, 2, 0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 1, 0, 0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 0, 1, 0, 1, 2, 2, 1, 0, 0, 1, 2, 2, 0, 1, 0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 1, 0, 2, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

This is the fixed point of the morphism 0->012, 1->201, 2->210 starting with 0. Let u be the (nonperiodic) sequence of positions of 0, and likewise, v for 1 and w for 2; then u(n)/n -> 3, v(n)/n -> 3, w(n)/n -> 3.

See A287385 for a guide to related sequences.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000

Index entries for sequences that are fixed points of mappings

EXAMPLE

First three iterations of the morphism: 012, 012201210, 012201210210012201210201012.

MATHEMATICA

s = Nest[Flatten[# /. {0->{0, 1, 2}, 1->{2, 0, 1}, 2->{2, 1, 0}}] &, {0}, 9]; (*A287455*)

Flatten[Position[s, 0]]; (*A287456*)

Flatten[Position[s, 1]]; (*A287457*)

Flatten[Position[s, 2]]; (*A287458*)

CROSSREFS

Cf. A287385, A287456, A287457, A287458.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 30 2017

STATUS

approved

A287456

Positions of 0 in A287455.

+10
4

1, 5, 9, 12, 13, 17, 21, 23, 25, 30, 32, 34, 37, 41, 45, 48, 49, 53, 57, 59, 61, 66, 67, 71, 73, 77, 81, 84, 86, 88, 93, 94, 98, 100, 104, 108, 109, 113, 117, 120, 121, 125, 129, 131, 133, 138, 140, 142, 145, 149, 153, 156, 157, 161, 165, 167, 169, 174, 175

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

a(n) - a(n-1) is in {1, 2, 3, 4, 5} for n >= 1; also, 3n - a(n) is in {0, 1, 2} for n >= 1. The first 20 numbers 3n - a(n) are 2, 1, 0, 0, 2, 1, 0, 1, 2, 0, 1, 2, 2, 1, 0, 0, 2, 1, 0, 1, with 0 in positions given by A287452, 1 in positions given by A287454, and 2 in positions given by A287458.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000

MATHEMATICA

s = Nest[Flatten[# /. {0->{0, 1, 2}, 1->{2, 0, 1}, 2->{2, 1, 0}}] &, {0}, 9]; (*A287455*)

Flatten[Position[s, 0]]; (*A287456*)

Flatten[Position[s, 1]]; (*A189666*)

Flatten[Position[s, 2]]; (*A287458*)

CROSSREFS

Cf. A287455, A189666, A287458.

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 30 2017

STATUS

approved

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