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%I #4 Jan 03 2020 20:19:12
%S 1,3,4,6,7,9,10,12,13,15,16,18,19,21,22,24,26,27,29,30,32,33,35,36,38,
%T 39,41,42,44,45,47,48,50,52,53,55,56,58,59,61,62,64,65,67,68,70,71,73,
%U 75,76,78,79,81,82,84,85,87,88,90,91,93,94,96,97,99,101
%N Beatty sequence for 2 + cos 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 + cos 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, A329961 (complement).
%K nonn,easy
%O 1,2
%A _Clark Kimberling_, Jan 02 2020
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