A329961 Beatty sequence for 2 + sin x, where x = least positive solution of 1/(2 + sin x) + 1/(2 + cos x) = 1.

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

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..50.

Eric Weisstein's World of Mathematics, Beatty Sequence.

Index entries for sequences related to Beatty sequences

Formula

a(n) = floor(n*(2 + sin x)), where x = 2.058943... is the constant in A329960.

Mathematica

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}]

Crossrefs

Cf. A329825, A329960, A329962 (complement).

Sequence in context: A190336 A276889 A276877 * A078608 A329988 A189934

Adjacent sequences: A329958 A329959 A329960 * A329962 A329963 A329964

Keyword

nonn,easy

Author

Clark Kimberling, Jan 02 2020

Status

approved