A003512 A Beatty sequence: floor(n*(sqrt(3) + 2)). (Formerly M2622)

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

1,1

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

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)

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 *)

Prog

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

Crossrefs

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

Sequence in context: A343028 A000572 A059568 * A246170 A190694 A310206

Adjacent sequences: A003509 A003510 A003511 * A003513 A003514 A003515

Keyword

nonn

Author

N. J. A. Sloane

Status

approved