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A286751 Positions of 1 in A286749; complement of A286750. 3
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 (list; graph; refs; listen; history; text; internal format)
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: A039015 A037453 A014528 * A293278 A087791 A334880

Adjacent sequences:  A286748 A286749 A286750 * A286752 A286753 A286754

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 14 2017

STATUS

approved

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Last modified July 24 01:41 EDT 2021. Contains 346269 sequences. (Running on oeis4.)