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A329939 Beatty sequence for cosh x, where csch x + sech x = 1 .

%I A329939 #5 Feb 16 2025 08:33:58
%S A329939 2,4,6,8,10,12,14,17,19,21,23,25,27,29,31,34,36,38,40,42,44,46,49,51,
%T A329939 53,55,57,59,61,63,66,68,70,72,74,76,78,81,83,85,87,89,91,93,95,98,
%U A329939 100,102,104,106,108,110,113,115,117,119,121,123,125,127,130,132
%N A329939 Beatty sequence for cosh x,  where csch x + sech x = 1 .
%C A329939 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 A329939 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BeattySequence.html">Beatty Sequence.</a>
%H A329939 <a href="/index/Be#Beatty">Index entries for sequences related to Beatty sequences</a>
%F A329939 a(n) = floor(n*cosh x), where x = 1.390148... is the constant in A329937; a(n) first differs from A329832(n) at n = 68.
%t A329939 Solve[1/Sinh[x] + 1/Cosh[x] == 1, x]
%t A329939 r = ArcSech[1/8 (4 - 4 Sqrt[2] - 9 Sqrt[5 + 4 Sqrt[2]] + (5 + 4 Sqrt[2])^(3/2))];
%t A329939 u = N[r, 250]
%t A329939 v = RealDigits[u][[1]];
%t A329939 Table[Floor[n*Sinh[r]], {n, 1, 150}]  (* A329938 *)
%t A329939 Table[Floor[n*Cosh[r]], {n, 1, 150}]  (* A329939 *)
%Y A329939 Cf. A329825, A329832, A329937, A329938 (complement).
%K A329939 nonn,easy
%O A329939 1,1
%A A329939 _Clark Kimberling_, Jan 02 2020