OFFSET
1,2
COMMENTS
Conjecture: 2 < n*r - a(n) < 6 for n >= 1, where r = 2 + sqrt(2).
Let (d(n)) be the sequence of first differences of (a(n)), i.e., d(n)=a(n+1)-a(n) for n=1,2,... Let sigma be the morphism fixing (x(n)), where x = A284893. Then x is a concatenation of words B(i), where B(i) is either sigma(0)=01 or sigma(1)=0111. The crucial observation is that d(n+1)=2 if and only if B(n)=01, and d(n+1) =4 if and only if B(n)=0111. It follows that (d(n)) = 2,4,2,4,4,4,... is the unique fixed point of the morphism tau: 2->24, 4->2444 (tau equals sigma up to a change of alphabet). - Michel Dekking, Jan 16 2018
Since the positions of 0 in x are partial sums of the terms of d, one can prove, using the Perron Frobenius theorem, that a weak form of the conjecture above holds: the sequence (n*r - a(n)) is bounded. This result can also be derived from the simple part (the second eigenvalue 2-sqrt(2) of the incidence matrix of the morphism tau: 2->24, 4->2444 is smaller than 1) of Theorem 1 in the paper "Symbolic discrepancy and self-similar dynamics" by Boris Adamczewski. - Michel Dekking, Jan 16 2018
LINKS
Clark Kimberling, Table of n, a(n) for n = 1..10000
B. Adamczewski, Symbolic discrepancy and self-similar dynamics, Annales de l'Institut Fourier 54 (2004), 2201-2234.
EXAMPLE
As a word, A284893 = 010111010..., in which 0 is in positions 1,3,7,9,...
MATHEMATICA
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Apr 16 2017
STATUS
approved