A286048 Positions of 1 in A286046; complement of A286047.

1, 3, 5, 8, 9, 11, 14, 15, 17, 19, 21, 24, 25, 27, 29, 32, 33, 35, 37, 40, 41, 43, 46, 47, 49, 51, 54, 55, 57, 59, 62, 63, 65, 67, 69, 72, 73, 75, 78, 79, 81, 83, 85, 88, 89, 91, 93, 96, 97, 99, 101, 104, 105, 107, 109, 111, 114, 115, 117, 120, 121, 123, 125

Offset

1,2

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)) = 2,2,3,1,2,3,1,... 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 1's are given by a decoration:

A->22, B->31, C->231, D->2.

This is the same decoration as used for A286047, but with the letters 3 and 1 interchanged. It follows directly that (d(n)) can be obtained as a letter to letter image of a morphic sequence, fixed point of 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->3, d->1, e->3, 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 1 is in positions 1,3,5,8,9,...

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* Thue-Morse, A010060 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"011" -> "1"}]
st = ToCharacterCode[w1] - 48 (* A286046 *)
Flatten[Position[st, 0]]  (* A286047 *)
Flatten[Position[st, 1]]  (* A286048 *)

Crossrefs

Cf. A010060, A286046, A286047.

Sequence in context: A047624 A108594 A325425 * A350460 A026223 A034784

Adjacent sequences: A286045 A286046 A286047 * A286049 A286050 A286051

Keyword

nonn,easy

Author

Clark Kimberling, May 07 2017

Status

approved