A022838 Beatty sequence for sqrt(3); complement of A054406.

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 38, 39, 41, 43, 45, 46, 48, 50, 51, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 71, 72, 74, 76, 77, 79, 81, 83, 84, 86, 88, 90, 91, 93, 95, 96, 98, 100, 102, 103, 105, 107, 109, 110, 112

Offset: 1

Author

Clark Kimberling

Keywords

• nonn

Comments

0 <= A144077(n) - a(n) <= 1. - Reinhard Zumkeller, Sep 09 2008

From Reinhard Zumkeller, Jan 20 2010: (Start)

A080757(n) = a(n+1) - a(n).

A171970(n) = floor(a(n)/2).

A171972(n) = a(A000290(n)). (End)

Numbers k>0 such that A194979(k+1) = A194979(k) + 1. - Clark Kimberling, Dec 02 2014

Powers of 2 (i.e, 1, 8, 32, 64, 128, 256, 512, 4096, 8192,...) appear at n=1, 5, 19, 37, 74, 148, 296, 2365, 4730, 18919, 75675, 151349, 302698, 605396, ... related to A293328. - R. J. Mathar, Jan 17 2025

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Clark Kimberling, Beatty sequences and trigonometric functions, Integers 16 (2016), #A15.
• Eric Weisstein's World of Mathematics, Beatty Sequence.
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A080757 (first differences), A194106 (partial sums), A194028 (even bisection), A184796 (prime terms).

Cf. A026255, A054406 (complement).

Programs

Haskell

a022838 = floor . (* sqrt 3) . fromIntegral
-- Reinhard Zumkeller, Sep 14 2014

Magma

[Floor(n*Sqrt(3)): n in [1..60]]; // G. C. Greubel, Sep 28 2018

Maple

A022838 := proc(n)
    floor(n*sqrt(3)) ;
end proc: # R. J. Mathar, Mar 25 2013

Mathematica

Table[Floor[n 3^(1/2)] , {n, 1, 65}] (* Geoffrey Critzer, Jan 11 2015 *)

PARI

vector(60, n, floor(n*sqrt(3))) \\ G. C. Greubel, Sep 28 2018

PARI

a(n)=sqrtint(3*n^2) \\ Charles R Greathouse IV, Nov 01 2021

Python

from math import isqrt
def A022838(n): return isqrt(3*n*n) # Chai Wah Wu, Aug 06 2022

Formula

a(n) = floor(n*sqrt(3)). - Reinhard Zumkeller, Jan 20 2010

a(n) = 2 * floor(n * (sqrt(3) - 1)) + floor(n * (2 - sqrt(3))) + 1. - Miko Labalan, Dec 03 2016