A330000 Beatty sequence for sqrt(x+1), where 1/sqrt(x-1) + 1/sqrt(x+1) = 1.

2, 4, 6, 9, 11, 13, 15, 18, 20, 22, 25, 27, 29, 31, 34, 36, 38, 40, 43, 45, 47, 50, 52, 54, 56, 59, 61, 63, 66, 68, 70, 72, 75, 77, 79, 81, 84, 86, 88, 91, 93, 95, 97, 100, 102, 104, 106, 109, 111, 113, 116, 118, 120, 122, 125, 127, 129, 132, 134, 136, 138

Offset

1,1

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..61.

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, A329999 (complement).

Sequence in context: A085148 A252169 A187842 * A059566 A329826 A330908

Adjacent sequences: A329997 A329998 A329999 * A330001 A330002 A330003

Keyword

nonn,easy

Author

Clark Kimberling, Jan 03 2020

Status

approved