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A287726 Positions of 0 in A287725; complement of A287727. 5
2, 7, 11, 16, 21, 25, 30, 34, 39, 44, 48, 53, 58, 62, 67, 71, 76, 81, 85, 90, 94, 99, 104, 108, 113, 118, 122, 127, 131, 136, 141, 145, 150, 155, 159, 164, 168, 173, 178, 182, 187, 191, 196, 201, 205, 210, 215, 219, 224, 228, 233, 238, 242, 247, 251 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: 2 < n*r - a(n) < 3 for n >= 1, where r = (7 + sqrt(5))/2.

From Michel Dekking, Feb 12 2021: (Start)

Let T be the transform given by T(0) = 1, T(1) = 011 that defines A287725.

The Fibonacci word A003849 is a fixed point of the morphism  sigma: 0->01, 1->0, and therefore also of the morphism sigma^2: 0->010, 1->01.

Now note that T(sigma^2(0)) =  T(010) = 10111,  T(sigma^2(1)) = T(01) = 1011.

We see from this that the sequence (a(n+1)-a(n)) of first differences 5,4,5,5,4,5,4,5,5,4,5,5,4..., of (a(n)) is a sequence on the two letters 4 and 5, and that in fact these two letters occur as the Fibonacci word on the alphabet {5,4}.

From Lemma 8 in the paper by Allouche and Dekking it follows that (a(n)) is the generalized Beatty sequence given by a(n) = floor(n*phi) + 3n - 2.

This immediately implies Kimberling's conjecture.

Note that r = (7 + sqrt(5))/2 = phi + 3, where phi is the golden mean. So

  n*r - a(n) =  n(phi+3) - floor(n*phi) + 3n - 2 = n*phi - floor(n*phi) - 2,

which lies in (2,3).

These bounds are best possible since the fractional part of n*phi is equidistributed modulo 1.

(End)

LINKS

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

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

J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, Moscow J. Comb. Number Th. 8, 325-341, 2019.

FORMULA

a(n) = floor(n*phi) + 3n - 2. - Michel Dekking, Feb 12 2021

MATHEMATICA

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

w = StringJoin[Map[ToString, s]]

w1 = StringReplace[w, {"0" -> "1", "1" -> "011"}]

st = ToCharacterCode[w1] - 48    (* A287725 *)

Flatten[Position[st, 0]]  (* A287726 *)

Flatten[Position[st, 1]]  (* A287727 *)

CROSSREFS

Cf. A287725, A287727.

Sequence in context: A140491 A122823 A063289 * A190518 A190750 A049635

Adjacent sequences:  A287723 A287724 A287725 * A287727 A287728 A287729

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Jun 02 2017

STATUS

approved

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Last modified May 15 20:18 EDT 2021. Contains 343920 sequences. (Running on oeis4.)