A285669 Positions of 0 in A285668; complement of A285670.

3, 7, 13, 17, 21, 27, 31, 35, 39, 43, 49, 53, 57, 63, 67, 71, 77, 81, 85, 89, 93, 99, 103, 107, 113, 117, 121, 127, 131, 135, 141, 145, 149, 155, 159, 163, 167, 171, 177, 181, 185, 191, 195, 199, 205, 209, 213, 217, 221, 227, 231, 235, 241, 245, 249, 255

Offset

1,1

Comments

Conjecture: -1 < n*r - a(n) < 4 for n>=1, where r = (5 + sqrt(17))/2.

From Michel Dekking, Feb 28 2021: (Start)

Here is a proof of a weak form of the conjecture by Kimberling.

The limit of a(n)/n as n tends to infinity is equal to the inverse 1/f0 of the frequency f0 of the letter 0 in the sequence (a(n)) (see also the argument in A285401).

From the Perron-Frobenius theorem we know that the frequency vector (f0,f1) is the normalized eigenvector associated to the Perron-Frobenius eigenvalue lambda of the incidence matrix M of the generating morphism 0->11, 1->1101.

Here,

M = |0 1|

|2 3|.

The Perron eigenvalue of M is the number lambda = (3+sqrt(17))/2, and [u,v]:=[1,(3+sqrt(17))/2] is an eigenvector with lambda. So

1/f0 = (u+v)/u = (5+sqrt(17))/2,

which identifies the number r in the conjecture.

The fact that (n*r - a(n)) is a bounded sequence can be proved using a general symbolic discrepancy result, Theorem 1 in Adamscewski's 2004 paper.

(End)

Links

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

Boris Adamczewski, Symbolic discrepancy and self-similar dynamics, Annales de l'Institut Fourier 54 (2004), 2201-2234.

Example

As a word, A285668 = 11011101..., in which 0 is in positions 3,7,13,17,...

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 1}, 1 -> {1, 1, 0, 1}}] &, {0}, 9]  (* A285668 *)
Flatten[Position[s, 0]]  (* A285669 *)
Flatten[Position[s, 1]]  (* A285670 *)

Crossrefs

Cf. A285668, A285670.

Sequence in context: A038923 A310254 A310255 * A063239 A063226 A288709

Adjacent sequences: A285666 A285667 A285668 * A285670 A285671 A285672

Keyword

nonn,easy

Author

Clark Kimberling, May 02 2017

Status

approved