A330000 - OEIS

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

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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