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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, May 02 2017
STATUS
approved