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A329938 Beatty sequence for sinh x, where csch x + sech x = 1 .

%I A329938 #6 Feb 16 2025 08:33:58
%S A329938 1,3,5,7,9,11,13,15,16,18,20,22,24,26,28,30,32,33,35,37,39,41,43,45,
%T A329938 47,48,50,52,54,56,58,60,62,64,65,67,69,71,73,75,77,79,80,82,84,86,88,
%U A329938 90,92,94,96,97,99,101,103,105,107,109,111,112,114,116,118
%N A329938 Beatty sequence for sinh x, where csch x + sech x = 1 .
%C A329938 Let x be the solution of csch x + sech x = 1. Then (floor(n*sinh x)) and (floor(n*cosh 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 A329938 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence.</a>
%H A329938 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%F A329938 a(n) = floor(n*sinh x), where x = 1.390148... is the constant in A329937; a(n) first differs from A329831(n) at n = 77.
%t A329938 Solve[1/Sinh[x] + 1/Cosh[x] == 1, x]
%t A329938 r = ArcSech[1/8 (4 - 4 Sqrt[2] - 9 Sqrt[5 + 4 Sqrt[2]] + (5 + 4 Sqrt[2])^(3/2))];
%t A329938 u = N[r, 250]
%t A329938 v = RealDigits[u][[1]];
%t A329938 Table[Floor[n*Sinh[r]], {n, 1, 150}]  (* A329938 *)
%t A329938 Table[Floor[n*Cosh[r]], {n, 1, 150}]  (* A329939 *)
%Y A329938 Cf. A329825, A329831, A329937, A329939 (complement).
%K A329938 nonn,easy
%O A329938 1,2
%A A329938 _Clark Kimberling_, Jan 02 2020