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

%I #4 Jan 04 2020 12:57:51

%S 2,5,7,10,12,15,17,20,22,25,27,30,32,35,37,40,42,45,47,50,53,55,58,60,

%T 63,65,68,70,73,75,78,80,83,85,88,90,93,95,98,100,103,106,108,111,113,

%U 116,118,121,123,126,128,131,133,136,138,141,143,146,148,151

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

%C Let x be the solution of 1/x + csch(x) = 1. Then (floor(n x) and (floor(n sinh(x))) 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 sinh(x)), where x = 1.656135560... is the constant in A330065.

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

%t RealDigits[r][[1]] (* A330065 *)

%t Table[Floor[n*r], {n, 1, 250}] (* A330066 *)

%t Table[Floor[n*Sinh[r]], {n, 1, 250}] (* A330067 *)

%Y Cf. A329825, A330065, A330066 (complement).

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Jan 04 2020