A003512 A Beatty sequence: floor(n*(sqrt(3) + 2)).

3, 7, 11, 14, 18, 22, 26, 29, 33, 37, 41, 44, 48, 52, 55, 59, 63, 67, 70, 74, 78, 82, 85, 89, 93, 97, 100, 104, 108, 111, 115, 119, 123, 126, 130, 134, 138, 141, 145, 149, 153, 156, 160, 164, 167, 171, 175, 179, 182, 186

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Aviezri S. Fraenkel, Jonathan Levitt, Michael Shimshoni, Characterization of the set of values f(n)=[n alpha], n=1,2,..., Discrete Math. 2 (1972), no.4, 335-345.
• Eric Weisstein's World of Mathematics, Beatty Sequence.
• Index entries for sequences related to Beatty sequences

Crossrefs

Cf. A003511 (complement), A019973 (sqrt(3)+2).

Programs

Maple

Digits := 60: A003512 := proc(n) trunc( evalf( n*(sqrt(3)+2) )); end;

Mathematica

Table[Floor[n (Sqrt@ 3 + 2)], {n, 50}] (* Michael De Vlieger, Oct 08 2016 *)

Python

from gmpy2 import isqrt
def A003512(n):
    return 2*n + int(isqrt(3*n**2))  # Chai Wah Wu, Oct 08 2016

Formula

a(n) = floor(n*(sqrt(3)+2)). - Michel Marcus, Jan 05 2015

For n >= 0, a(n) = 2n + largest integer m such that m^2 <= 3*n^2. - Chai Wah Wu, Oct 08 2016

From Miko Labalan, Dec 03 2016: (Start)

For n > 0, a(n) = 4*floor(n*(sqrt(3)-1)) + 3*floor(n*(2-sqrt(3))) + 3;

a(0) = 0, a(n) = a(n - 1) + A182778(n) - A182778(n - 1) - 1.

(End)