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Beatty sequence for 2 + sin x, where x = least positive solution of 1/(2 + sin x) + 1/(2 + cos x) = 1.
3

%I #4 Jan 03 2020 20:19:04

%S 2,5,8,11,14,17,20,23,25,28,31,34,37,40,43,46,49,51,54,57,60,63,66,69,

%T 72,74,77,80,83,86,89,92,95,98,100,103,106,109,112,115,118,121,123,

%U 126,129,132,135,138,141,144

%N Beatty sequence for 2 + sin x, where x = least positive solution of 1/(2 + sin x) + 1/(2 + cos x) = 1.

%C Let x be the least positive solution of 1/(2 + sin x) + 1/(2 + cos x) = 1. Then (floor(n*(2 + sin x))) and (floor(n*(2 + cos x))) 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*(2 + sin x)), where x = 2.058943... is the constant in A329960.

%t Solve[1/(2 + Sin[x]) + 1/(2 + Cos[x]) == 1, x]

%t u = ArcCos[-(1/2) + 1/Sqrt[2] - 1/2 Sqrt[-1 + 2 Sqrt[2]]]

%t u1 = N[u, 150]

%t RealDigits[u1, 10][[1]] (* A329960 *)

%t Table[Floor[n*(2 + Sin[u])], {n, 1, 50}] (* A329961 *)

%t Table[Floor[n*(2 + Cos[u])], {n, 1, 50}] (* A329962 *)

%t Plot[1/(2 + Sin[x]) + 1/(2 + Cos[x]) - 1, {x, -1, 3}]

%Y Cf. A329825, A329960, A329962 (complement).

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jan 02 2020