A286047 - OEIS

A286047

Positions of 0 in A286046; complement of A286048.

2, 4, 6, 7, 10, 12, 13, 16, 18, 20, 22, 23, 26, 28, 30, 31, 34, 36, 38, 39, 42, 44, 45, 48, 50, 52, 53, 56, 58, 60, 61, 64, 66, 68, 70, 71, 74, 76, 77, 80, 82, 84, 86, 87, 90, 92, 94, 95, 98, 100, 102, 103, 106, 108, 110, 112, 113, 116, 118, 119, 122, 124

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OFFSET

1,1

COMMENTS

Conjecture: 2n - a(n) is in {0,1} for n >= 1.

From Michel Dekking, Apr 12 2022: (Start)

Obviously Kimberling's conjecture is equivalent to the property that A286046 is a concatenation of the two 2-blocks 01 and 10. This can be read off immediately from the {A, B, C, D} composition of A286046 given in the comments of that sequence.

But more is true. The first difference sequence (d(n)) = 2,2,1,3,2,1,3,... of (a(n)) is a morphic sequence. From the representation of A286046 by the decoration A->1010, B->1001, C->101001, D->10, we see that the differences between occurrences of 0's are given by a decoration:

A->22, B->13, C->213, D->2.

The 'natural' algorithm to obtain (d(n)) as a letter to letter image of a morphic sequence from this decoration yields (for example) a morphism mu on an alphabet {a,b,c,d,e,f} given by

mu: a->ab, b->cd, c->aed, d->f, e->cd, f->aed,

with the letter-to-letter map

lambda: a->2, b->2, c->1, d->3, e->1, f->2.

We have (d(n)) = lambda(z), where z is the fixed point z = abcdae... of mu.

(End)

LINKS

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

EXAMPLE

As a word, A286046 = 101010011010011010..., in which 0 is in positions 2,4,6,7,10,...

MATHEMATICA

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* Thue-Morse, A010060 *)

w = StringJoin[Map[ToString, s]]