A158919 Beatty sequence for the tribonacci constant tau (A058265): a(n) = floor(n*tau).

0, 1, 3, 5, 7, 9, 11, 12, 14, 16, 18, 20, 22, 23, 25, 27, 29, 31, 33, 34, 36, 38, 40, 42, 44, 45, 47, 49, 51, 53, 55, 57, 58, 60, 62, 64, 66, 68, 69, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 90, 91, 93, 95, 97, 99, 101, 103, 104, 106, 108, 110

Offset: 0

Author

Eric Culver (weux082690(AT)yahoo.com), Mar 30 2009

Keywords

• nonn

Comments

Also called the spectrum of tau. Note that A276384 agrees with this sequence for n <= 17160 but disagrees beyond that point. In fact a(17161) = 31564, whereas A276384(17161) = 31563. - N. J. A. Sloane, Sep 03 2016

Examples

			a(3) = floor(3*q) = floor(3*1.8392867...) = floor(5.51786...) = 5.
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..20000
• A. J. Hildebrand, Junxian Li, Xiaomin Li, Yun Xie, Almost Beatty Partitions, arXiv:1809.08690 [math.NT], 2018.
• Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
• N. J. A. Sloane, Table of n, a(n) for n = 0..100000
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A058265, A140099 (spectrum of 1+tau), A276384, A277722, A277723.

Programs

Magma

[Floor(n * (1/3 + 1/3 * (19 - 3 * Sqrt(33))^(1/3) + 1/3 * (19 + 3 * Sqrt(33))^(1/3))) : n in [0..80]]; // Vincenzo Librandi, Oct 28 2018

Maple

x := (1/3)*(1+(19+3*sqrt(33))^(1/3)+(19-3*sqrt(33))^(1/3)) ;
[seq(floor(n*x),n=0..200)]; # R. J. Mathar, Sep 11 2011

Mathematica

a[n_] := Floor[n Root[#^3 - #^2 - # - 1&, 1]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 28 2018 *)

Formula

a(n) = floor(n*A058265).