A329999 Beatty sequence for sqrt(x-1), where 1/sqrt(x-1) + 1/sqrt(x+1) = 1.

1, 3, 5, 7, 8, 10, 12, 14, 16, 17, 19, 21, 23, 24, 26, 28, 30, 32, 33, 35, 37, 39, 41, 42, 44, 46, 48, 49, 51, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 71, 73, 74, 76, 78, 80, 82, 83, 85, 87, 89, 90, 92, 94, 96, 98, 99, 101, 103, 105, 107, 108, 110, 112, 114

Offset

1,2

Comments

Let x be the solution of 1/sqrt(x-1) + 1/sqrt(x+1) = 1. Then (floor(n sqrt(x-1))) and (floor(n sqrt(x+1))) 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..64.

Eric Weisstein's World of Mathematics, Beatty Sequence.

Index entries for sequences related to Beatty sequences

Formula

a(n) = floor(n sqrt(x-1)), where x = 4.18112544... is the constant in A329998.

Mathematica

r = x /. FindRoot[1/Sqrt[x - 1] + 1/Sqrt[x + 1] == 1, {x, 2, 10}, WorkingPrecision -> 120]
RealDigits[r][[1]] (* A329998 *)
Table[Floor[n*Sqrt[r - 1]], {n, 1, 250}]  (* A329999 *)
Table[Floor[n*Sqrt[r + 1]], {n, 1, 250}]  (* A330000 *)

Crossrefs

Cf. A329825, A329998, A330000 (complement).

Sequence in context: A327212 A329825 A059565 * A187841 A253301 A003144

Adjacent sequences: A329996 A329997 A329998 * A330000 A330001 A330002

Keyword

nonn,easy

Author

Clark Kimberling, Jan 03 2020

Status

approved