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Beatty sequence for (-1+sqrt(41))/4.
3

%I #5 Jan 02 2020 08:22:54

%S 1,2,4,5,6,8,9,10,12,13,14,16,17,18,20,21,22,24,25,27,28,29,31,32,33,

%T 35,36,37,39,40,41,43,44,45,47,48,49,51,52,54,55,56,58,59,60,62,63,64,

%U 66,67,68,70,71,72,74,75,76,78,79,81,82,83,85,86,87,89

%N Beatty sequence for (-1+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 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 = (-1+sqrt(41))/5.

%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, A329840 (complement).

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Dec 31 2019