A330000 - OEIS

%I #4 Jan 04 2020 12:56:33

%S 2,4,6,9,11,13,15,18,20,22,25,27,29,31,34,36,38,40,43,45,47,50,52,54,

%T 56,59,61,63,66,68,70,72,75,77,79,81,84,86,88,91,93,95,97,100,102,104,

%U 106,109,111,113,116,118,120,122,125,127,129,132,134,136,138

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

%C 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.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence.</a>

%H <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>

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

%t r = x /. FindRoot[1/Sqrt[x - 1] + 1/Sqrt[x + 1] == 1, {x, 2, 10}, WorkingPrecision -> 120]

%t RealDigits[r][[1]] (* A329998 *)

%t Table[Floor[n*Sqrt[r - 1]], {n, 1, 250}]  (* A329999 *)

%t Table[Floor[n*Sqrt[r + 1]], {n, 1, 250}]  (* A330000 *)

%Y Cf. A329825, A329998, A329999 (complement).

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jan 03 2020