A141104 Lower Even Swappage of Upper Wythoff Sequence.

2, 4, 6, 10, 12, 14, 18, 20, 22, 26, 28, 30, 34, 36, 38, 40, 44, 46, 48, 52, 54, 56, 60, 62, 64, 68, 70, 72, 74, 78, 80, 82, 86, 88, 90, 94, 96, 98, 102, 104, 106, 108, 112, 114, 116, 120, 122, 124, 128, 130, 132, 136, 138, 140, 142, 146, 148, 150, 154, 156, 158, 162

Offset

1,1

Comments

lim (1/n)*A141104(n) = 1 + tau.

Let S(n)=(1/2)*A141104(n). Is the complement of S equal to A004976?

This question has an affirmative answer, as proved by Russo and Schwiebert in the link below. It can also be proved using the Walnut theorem-prover, using synchronized Fibonacci automata for the two sequences. 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

Table of n, a(n) for n=1..62.

Vincent Russo and Loren Schwiebert, Beatty Sequences, Fibonacci Numbers, and the Golden Ratio, The Fibonacci Quarterly, Vol 49, Number 2, May 2011.

Luke Schaeffer, Jeffrey Shallit, and Stefan Zorcic, Beatty Sequences for a Quadratic Irrational: Decidability and Applications, arXiv:2402.08331 [math.NT], 2024. See p. 14.

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 odd b(n), let a(m) be the least number in a such that after swapping b(n) and a(m), the resulting new a and b are both increasing. A141104 is the sequence obtained by thus swapping all odds 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 = (1,3,5,6,8,9,11,12,...) and b = (2,4,7,10,13,15,18,...).
After 2nd swap,
a = (1,3,5,7,8,9,11,12,...) and b = (2,4,6,10,13,15,18,...).

Crossrefs

Cf. A000201, A001950, A141105, A141106, A141107, A004976.

Sequence in context: A103799 A175817 A256773 * A047410 A339331 A255056

Adjacent sequences: A141101 A141102 A141103 * A141105 A141106 A141107

Keyword

nonn

Author

Clark Kimberling, Jun 02 2008, Aug 27 2008

Status

approved