A286751 Positions of 1 in A286749; complement of A286750.

1, 2, 5, 6, 7, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 28, 29, 30, 33, 34, 37, 38, 39, 42, 43, 44, 47, 48, 51, 52, 53, 56, 57, 58, 61, 62, 65, 66, 67, 70, 71, 74, 75, 76, 79, 80, 81, 84, 85, 88, 89, 90, 93, 94, 97, 98, 99, 102, 103, 104, 107, 108, 111, 112

Offset

1,2

Comments

a(n) - a(n-1) is in {1,2,3} for n>=2, and a(n)/n -> 4 - sqrt(5).

From Michel Dekking, Aug 29 2020: (Start)

This sequence is a generalized Beatty sequence.

Recall from A286749 that A286749 is the letter-to-letter image of the fixed point x of the morphism mu given by

mu: 1->12341, 2->1, 3->2, 4->34.

where the letter-to-letter map lambda is defined by

lambda: 1->1, 2->1, 3->0, 4->0.

The return words of the word 1 in x are A:=1 and B:=1234.

We have mu(1)=12341, and mu(1234)=123411234. So the derived morphism is tau: A->BA, B->BAB.

This morphism happens to be the square of the Fibonacci morphism on the alphabet {B,A}.

The return word A = 1 has lambda-image 1, and the return word B = 1234 has lambda-image 1100. This means that they give distances 1, respectively 1 and 3 between (successive) occurrences of 1's in A286749. This leads to the decoration B->13, A->1, which amounts to replacing 0 by 01 and 1 by 0 in the Fibonacci word.

But the Fibonacci word is fixed by the {0,1} version of tau. It follows that the sequence (a(n+1)-a(n)) = 1,3,1,1,3,... is the Fibonacci word on the alphabet {1,3}. Finally, Lemma 8 in the paper "Generalized Beatty sequences..." then gives that a(n) = 5*n-2*floor(n*phi)-2, for n>0.

Clearly this implies a(n)/n -> 4 - sqrt(5).

(End)

Links

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

J.-P. Allouche, F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018-2019.

Formula

a(n) = 5*n-2*floor(n*phi)-2. - Michel Dekking, Aug 29 2020

Example

As a word, A286749 = 11001110011001110011010..., in which 1 is in positions 1,2,5,6,7,10,...

Mathematica

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 12]; (* A003849 *)
w = StringJoin[Map[ToString, s]];
w1 = StringReplace[w, {"0100" -> ""}]; st = ToCharacterCode[w1] - 48; (* A286749 *)
Flatten[Position[st, 0]];  (* A286750 *)
Flatten[Position[st, 1]];  (* A286751 *)

Crossrefs

Cf. A003849, A286749, A286750.

Sequence in context: A037453 A014528 A359375 * A293278 A087791 A334880

Adjacent sequences: A286748 A286749 A286750 * A286752 A286753 A286754

Keyword

nonn,easy

Author

Clark Kimberling, May 14 2017

Status

approved