cp's OEIS Frontend

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, 1, 0, 2, 1, 2, 0, 2, 1, 0, 1, 0, 2, 1, 2, with 0's in positions given by A189630 (conjectured), 1's in positions given by A287522 (conjectured), and 2's in positions given by A287521 (conjectured). [Corrected by Michel Dekking, Sep 30 2019]

From Michel Dekking, Oct 02 2019: (Start)

Proof of the conjecture about the positions of 2's in A287520.

The sequence A287520 =: x is, by definition, the fixed point starting with 0 of the morphism sigma given by

sigma(0) = 012, sigma(1) = 102, sigma(2) = 120.

The sequence A189630 is, by definition, the sequence of positions of 1's in A189628, where A189628 =: y is the unique fixed point of the morphism tau given by

tau(0) = 001, tau(1) = 010.

Let pi be the projection map from {0,1,2}* to {0,1}* given by

pi(0) = pi(1) = 0, pi(2) = 1.

Then pi sigma = tau pi on {0,1,2}*, and so pi(x) = y.

It follows directly that the positions of 2's in x are equal to the positions of 1's = pi(2) in y.

(End)

a(n) - a(n-1) is in {2, 3, 4} for n >= 1; also, 3n - a(n) is in {1, 2} for n >= 1. The first 20 numbers 3n - a(n) are 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 1, 2, 1, with 1 in positions given by A287521.

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