login
Beatty sequence for sinh(x), where 1/e^x + csch(x) = 1.
3

%I #4 Jan 05 2020 08:12:17

%S 1,2,4,5,7,8,10,11,13,14,15,17,18,20,21,23,24,26,27,29,30,31,33,34,36,

%T 37,39,40,42,43,44,46,47,49,50,52,53,55,56,58,59,60,62,63,65,66,68,69,

%U 71,72,74,75,76,78,79,81,82,84,85,87,88,89,91,92,94,95

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

%C Let x be the positive solution of 1/e^x + csch(x) = 1. Then (floor(n e^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 = 1.1676157... is the constant in A330115.

%t r = x /. FindRoot[1/E^x + Csch[x] == 1, {x, 1, 2}, WorkingPrecision -> 200]

%t RealDigits[r][[1]] (* A330114 *)

%t Table[Floor[n*E^r], {n, 1, 250}] (* A330115 *)

%t Table[Floor[n*Sinh[r]], {n, 1, 250}] (* A330116 *)

%Y Cf. A329825, A330114, A330115 (complement).

%K nonn,easy

%O 1,2

%A _Clark Kimberling_, Jan 04 2020