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

%I #12 Aug 27 2022 21:31:38

%S 1,3,4,6,8,9,11,12,14,16,17,19,21,22,24,25,27,29,30,32,33,35,37,38,40,

%T 42,43,45,46,48,50,51,53,54,56,58,59,61,63,64,66,67,69,71,72,74,76,77,

%U 79,80,82,84,85,87,88,90,92,93,95,97,98,100,101,103,105

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

%C Let x be the solution of 1/x + sech(x) = 1. Then floor(n*x) and floor(n*cosh(x)) define a pair of Beatty sequences indexed by n; i.e., every positive integer is in exactly one of the sequences. See the Guide to related sequences at A329825.

%C First differs from A000201 at a(34) = 54 <> 55. - _Peter Munn_, Aug 27 2022

%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*x), where x = 1.61749989... is the constant in A330062.

%t r = x /. FindRoot[1/x + 1/Cosh[x] == 1, {x, 2, 10}, WorkingPrecision -> 210]

%t RealDigits[r][[1]] (* A330062 *)

%t Table[Floor[n*r], {n, 1, 250}] (* A330063 *)

%t Table[Floor[n*Cosh[r]], {n, 1, 250}] (* A330064 *)

%Y Cf. A000201, A329825, A330062, A330064 (complement).

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jan 04 2020