A182777 Beatty sequence for 3-sqrt(3).

1, 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 38, 39, 40, 41, 43, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73, 74, 76, 77, 78, 79, 81, 82, 83, 84, 86

Offset: 1

Author

Clark Kimberling, Nov 30 2010

Keywords

• nonn

Comments

(1) 3 is the only number x for which the numbers r=x-sqrt(x) and s=x+sqrt(x) satisfy the Beatty equation

1/r + 1/s = 1.

(2) Let u=2-sqrt(3) and v=1. Jointly rank {j*u} and {k*v} as in the first comment at A182760; a(n) is the position of n*u.

(3) The complement of A182777 is A182778, which gives the positions of the natural numbers k in the joint ranking.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A182760, A182778.

Programs

Magma

[Floor(n*(3-Sqrt(3))): n in [1..80]]; // Vincenzo Librandi, Oct 25 2011

Mathematica

Table[Floor[(3-Sqrt[3]) n], {n, 68}]

PARI

vector(80, n, floor(n*(3-sqrt(3)))) \\ G. C. Greubel, Nov 23 2018

Sage

[floor(n*(3-sqrt(3))) for n in (1..80)] # G. C. Greubel, Nov 23 2018

Formula

a(n) = floor(n*(3-sqrt(3))).

Extensions

Typo in formula by Vincenzo Librandi, Oct 25 2011