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

%I #4 Jan 04 2020 12:57:44

%S 1,3,4,6,8,9,11,13,14,16,18,19,21,23,24,26,28,29,31,33,34,36,38,39,41,

%T 43,44,46,48,49,51,52,54,56,57,59,61,62,64,66,67,69,71,72,74,76,77,79,

%U 81,82,84,86,87,89,91,92,94,96,97,99,101,102,104,105,107

%N Beatty sequence for x, where 1/x + csch(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 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, A330067 (complement).

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jan 04 2020