login
Beatty sequence for (11+sqrt(101))/10.
3

%I #5 Feb 16 2025 08:33:58

%S 2,4,6,8,10,12,14,16,18,21,23,25,27,29,31,33,35,37,39,42,44,46,48,50,

%T 52,54,56,58,61,63,65,67,69,71,73,75,77,79,82,84,86,88,90,92,94,96,98,

%U 101,103,105,107,109,111,113,115,117,119,122,124,126,128,130

%N Beatty sequence for (11+sqrt(101))/10.

%C Let r = (9+sqrt(101))/10. Then (floor(n*r)) and (floor(n*r + 3r/4)) 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="https://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*s), where s = (11+sqrt(101))/10.

%t t = 1/5; r = Simplify[(2 - t + Sqrt[t^2 + 4])/2]; s = Simplify[r/(r - 1)];

%t Table[Floor[r*n], {n, 1, 200}] (* A329835 *)

%t Table[Floor[s*n], {n, 1, 200}] (* A329836 *)

%Y Cf. A329825, A329835 (complement).

%K nonn,easy,changed

%O 1,1

%A _Clark Kimberling_, Dec 31 2019