login
Beatty sequence for (2+sqrt(10))/3.
3

%I #4 Jan 02 2020 08:19:49

%S 1,3,5,6,8,10,12,13,15,17,18,20,22,24,25,27,29,30,32,34,36,37,39,41,

%T 43,44,46,48,49,51,53,55,56,58,60,61,63,65,67,68,70,72,73,75,77,79,80,

%U 82,84,86,87,89,91,92,94,96,98,99,101,103,104,106,108,110

%N Beatty sequence for (2+sqrt(10))/3.

%C Let r = (2+sqrt(10))/3. Then (floor(n*r)) and (floor(n*r + 2r/3)) are a pair of Beatty sequences; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

%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*r), where r = (2+sqrt(10))/3.

%t t = 2/3; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];

%t Table[Floor[r*n], {n, 1, 200}] (* A329829 *)

%t Table[Floor[s*n], {n, 1, 200}] (* A329830 *)

%Y Cf. A329825, A329830 (complement).

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Dec 31 2019