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A329961
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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.
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3
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2, 5, 8, 11, 14, 17, 20, 23, 25, 28, 31, 34, 37, 40, 43, 46, 49, 51, 54, 57, 60, 63, 66, 69, 72, 74, 77, 80, 83, 86, 89, 92, 95, 98, 100, 103, 106, 109, 112, 115, 118, 121, 123, 126, 129, 132, 135, 138, 141, 144
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = floor(n*(2 + sin x)), where x = 2.058943... is the constant in A329960.
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MATHEMATICA
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Solve[1/(2 + Sin[x]) + 1/(2 + Cos[x]) == 1, x]
u = ArcCos[-(1/2) + 1/Sqrt[2] - 1/2 Sqrt[-1 + 2 Sqrt[2]]]
u1 = N[u, 150]
RealDigits[u1, 10][[1]] (* A329960 *)
Table[Floor[n*(2 + Sin[u])], {n, 1, 50}] (* A329961 *)
Table[Floor[n*(2 + Cos[u])], {n, 1, 50}] (* A329962 *)
Plot[1/(2 + Sin[x]) + 1/(2 + Cos[x]) - 1, {x, -1, 3}]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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