login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A287665 Positions of 1's in A287663; complement of A287664. 3
1, 5, 6, 10, 14, 15, 19, 20, 24, 28, 29, 33, 37, 38, 42, 43, 47, 51, 52, 56, 57, 61, 65, 66, 70, 74, 75, 79, 80, 84, 88, 89, 93, 97, 98, 102, 103, 107, 111, 112, 116, 117, 121, 125, 126, 130, 134, 135, 139, 140, 144, 148, 149, 153, 154, 158, 162, 163, 167 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Conjecture: 0 < n*r - a(n) < 3 for n >= 1, where r = (-1 + 3*sqrt(5))/2. [Corrected by Clark Kimberling, Aug 19 2019]

From Michel Dekking, Aug 21 2019: (Start)

Proof of Maiga's conjecture: let T denote the morphism {0->1, 1->000}.

The Fibonacci word xF:=A003849 is fixed point of the morphism  0->01, 1->0 and therefore xF is a concatenation of the two words v=01 and w=0, where these words occur as the Fibonacci word itself.

Now note that

T(v) = T(01) = 1000, T(w) = T(0) = 1.

We see from this that the sequence of first differences of A287665, Delta A287665 = 4,1,4,4,1,4,1,4,4,1,..., is a sequence on the letters 4 and 1, and that in fact these two letters occur as the Fibonacci word on the alphabet {4,1}.

Since A001468 (starting from n=1) is the Fibonacci word on the alphabet {2,1}, Maiga's formula follows.

Proof of Kimberling's conjecture.

It follows from the result above by Lemma 8 in the Allouche-Dekking paper that A287665 is a generalized Beatty sequence

       a(n) = 3*floor(n*phi) - 2*n.

So if r = (-1 + 3*sqrt(5))/2 = 3*phi - 2, then

    n*r - a(n) = n*(3*phi-2) - [3*(n*phi-{n*phi})]-2*n = 3*{n*phi}, where {} denotes fractional part.

It follows that 0 < n*r - a(n) < 3. Moreover, these bounds are tight, since the sequence ({n*phi}) is equidistributed on (0,1).

(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.

FORMULA

a(n) = Sum_{k=0..n-1} [3*A001468(k)-2] (conjectured). - Jon Maiga, Dec 30 2018

a(n) = 3*floor(n*phi) - 2*n. - Michel Dekking, Aug 21 2019

MATHEMATICA

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 10] (* A003849 *)

w = StringJoin[Map[ToString, s]]

w1 = StringReplace[w, {"0" -> "1", "1" -> "000"}]

st = ToCharacterCode[w1] - 48    (* A287663 *)

Flatten[Position[st, 0]]  (* A287664 *)

Flatten[Position[st, 1]]  (* A287665 *)

CROSSREFS

Cf. A287663, A287664, A001468.

Sequence in context: A166563 A005847 A109758 * A015820 A096728 A011985

Adjacent sequences:  A287662 A287663 A287664 * A287666 A287667 A287668

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 02 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 20 04:37 EDT 2019. Contains 328247 sequences. (Running on oeis4.)