A287528 -id:A287528 - cp's OEIS frontend

A287529 Positions of 0 in A287528.

1, 5, 9, 11, 13, 18, 21, 23, 25, 29, 31, 36, 37, 41, 45, 48, 50, 52, 57, 59, 61, 65, 67, 72, 73, 77, 81, 83, 85, 90, 91, 95, 99, 102, 104, 106, 109, 113, 117, 119, 121, 126, 129, 131, 133, 138, 140, 142, 146, 148, 153, 154, 158, 162, 165, 167, 169, 173, 175

Offset: 1

Clark Kimberling, May 30 2017

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, 1, 2, 0, 0, 1, 2, 1, 2, 0, 2, 1, 0, 0, 1, 2, 0, 1, with 0 in positions given by A189634, 1 in positions given by A189670, and 2 in positions given by A189670.

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

Cf. A189670, A287528, A189634.

s = Nest[Flatten[# /. {0->{0, 1, 2}, 1->{1, 0, 2}, 2->{2, 1, 0}}] &, {0}, 9]; (*A287528*)
Flatten[Position[s, 0]]; (*A287529*)
Flatten[Position[s, 1]]; (*A189670*)
Flatten[Position[s, 2]]; (*A189634*)

A287385 Start with 0 and repeatedly substitute 0->012, 1->210, 2->021.

Original entry on oeis.org

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

Offset: 1

Clark Kimberling, May 25 2017

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

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

Clark Kimberling, Table of n, a(n) for n = 1..10000
Index entries for sequences that are fixed points of mappings

Cf. A287386, A287387, A287388.

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*)

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

cp's OEIS Frontend

A287529 Positions of 0 in A287528.

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