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Beatty sequence for sqrt(x-1), where 1/sqrt(x-1) + 1/sqrt(x+1) = 1.
3

%I #4 Jan 04 2020 12:56:21

%S 1,3,5,7,8,10,12,14,16,17,19,21,23,24,26,28,30,32,33,35,37,39,41,42,

%T 44,46,48,49,51,53,55,57,58,60,62,64,65,67,69,71,73,74,76,78,80,82,83,

%U 85,87,89,90,92,94,96,98,99,101,103,105,107,108,110,112,114

%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, A330000 (complement).

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jan 03 2020