A329962 Beatty sequence for 2 + cos x, where x = least positive solution of 1/(2 + sin x) + 1/(2 + cos x) = 1.

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, 39, 41, 42, 44, 45, 47, 48, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 101

Offset

1,2

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..66.

Eric Weisstein's World of Mathematics, Beatty Sequence.

Index entries for sequences related to Beatty sequences

Formula

a(n) = floor(n*(2 + cos 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, A329961 (complement).

Sequence in context: A032766 A189935 A329987 * A258574 A064717 A109231

Adjacent sequences: A329959 A329960 A329961 * A329963 A329964 A329965

Keyword

nonn,easy

Author

Clark Kimberling, Jan 02 2020

Status

approved