A287521 - OEIS

A287521

Positions of 0's in A287520.

1, 5, 9, 11, 13, 18, 20, 24, 25, 29, 31, 36, 37, 41, 45, 47, 51, 52, 56, 58, 63, 65, 69, 70, 73, 77, 81, 83, 85, 90, 91, 95, 99, 101, 105, 106, 109, 113, 117, 119, 121, 126, 128, 132, 133, 137, 139, 144, 146, 150, 151, 154, 158, 162, 164, 166, 171, 172, 176

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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, 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)

LINKS

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

MATHEMATICA

s = Nest[Flatten[# /. {0->{0, 1, 2}, 1->{1, 0, 2}, 2->{1, 2, 0}}] &, {0}, 9]; (*A287520*)

Flatten[Position[s, 0]]; (* A287521 *)

Flatten[Position[s, 1]]; (* A287522 *)