OFFSET
1,2
COMMENTS
This is one of three sequences that partition the positive integers. In general, suppose that r, s, t are positive real numbers for which the sets {i/r: i>=1}, {j/s: j>=1}, {k/t: k>=1} are pairwise disjoint. Let a(n) be the rank of n/r when all the numbers in the three sets are jointly ranked. Define b(n) and c(n) as the ranks of n/s and n/t. It is easy to prove that
a(n) = n+[ns/r]+[nt/r],
b(n) = n+[nr/s]+[nt/s],
c(n) = n+[nr/t]+[ns/t], where []=floor.
(Conjecture) These are the numbers n such that (n+1)-sections of the Fibonacci word contain "000" (the commoner bit) but not "111" (the rarer bit). - Don Reble, Apr 07 2021
Conjecture proved April 8 2021, using the Walnut theorem prover. - Jeffrey Shallit, Apr 09 2021
LINKS
Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See pp. 11-12.
MATHEMATICA
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Apr 20 2011
STATUS
approved