login
Beatty sequence for (9+sqrt(41))/4.
3

%I #14 Oct 16 2023 09:28:17

%S 3,7,11,15,19,23,26,30,34,38,42,46,50,53,57,61,65,69,73,77,80,84,88,

%T 92,96,100,103,107,111,115,119,123,127,130,134,138,142,146,150,154,

%U 157,161,165,169,173,177,180,184,188,192,196,200,204,207,211,215,219

%N Beatty sequence for (9+sqrt(41))/4.

%C Let r = (-1+sqrt(41))/4. Then (floor(n*r)) and (floor(n*r + 5r/2)) 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 Michael De Vlieger, <a href="/A329840/b329840.txt">Table of n, a(n) for n = 1..10000</a>

%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*s), where s = (9+sqrt(41))/4. - corrected by _Michael De Vlieger_, Aug 27 2021

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

%t Table[Floor[r*n], {n, 1, 200}] (* A329839 *)

%t Table[Floor[s*n], {n, 1, 200}] (* A329840 *)

%Y Cf. A329825, A329839 (complement).

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Dec 31 2019