%I M2622 #52 Dec 05 2016 11:47:05

%S 3,7,11,14,18,22,26,29,33,37,41,44,48,52,55,59,63,67,70,74,78,82,85,

%T 89,93,97,100,104,108,111,115,119,123,126,130,134,138,141,145,149,153,

%U 156,160,164,167,171,175,179,182,186

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

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

%H Chai Wah Wu, <a href="/A003512/b003512.txt">Table of n, a(n) for n = 1..10000</a>

%H Aviezri S. Fraenkel, Jonathan Levitt, Michael Shimshoni, <a href="http://dx.doi.org/10.1016/0012-365X(72)90012-X">Characterization of the set of values f(n)=[n alpha], n=1,2,...</a>, Discrete Math. 2 (1972), no.4, 335-345.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence.</a>

%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>

%F a(n) = floor(n*(sqrt(3)+2)). - _Michel Marcus_, Jan 05 2015

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

%F From _Miko Labalan_, Dec 03 2016: (Start)

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

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

%F (End)

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

%t Table[Floor[n (Sqrt@ 3 + 2)], {n, 50}] (* _Michael De Vlieger_, Oct 08 2016 *)

%o (Python)

%o from gmpy2 import isqrt

%o def A003512(n):

%o return 2*n + int(isqrt(3*n**2)) # _Chai Wah Wu_, Oct 08 2016

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

%K nonn

%O 1,1

%A _N. J. A. Sloane_