OFFSET
1,1
COMMENTS
1. lim (1/n)*A141107(n) = 1 + tau
Both #2 and #3 are true. They can be proved with the Walnut theorem-prover, using the synchronized Fibonacci automaton for the sequences A141104 and A141107. These automata take n and y as input, in Fibonacci (Zeckendorf) representation, and accept iff y = a(n) for the respective sequence. - Jeffrey Shallit, Jan 27 2024
LINKS
Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See p. 15.
FORMULA
Let a = (1,3,4,6,8,9,11,12,...) = A000201 = lower Wythoff sequence; let b = (2,5,7,10,13,15,18,...) = A001950 = upper Wythoff sequence. For each even b(n), let a(m) be the greatest number in a such that after swapping b(n) and a(m), the resulting new a and b are both increasing. A141107 is the sequence obtained by thus swapping all evens out of A001950.
EXAMPLE
Start with
a = (1,3,4,6,8,9,11,12,...) and b = (2,5,7,10,13,15,18,...).
After first swap,
a = (1,2,4,6,8,9,11,12,...) and b = (3,5,7,10,13,15,18,...).
After 2nd swap,
a = (1,2,4,6,8,9,10,12,...) and b = (3,5,7,11,13,15,18,...).
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jun 02 2008
STATUS
approved