A287726 Positions of 0 in A287725; complement of A287727.

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

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