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

2, 4, 6, 8, 10, 12, 14, 17, 19, 21, 23, 25, 27, 29, 31, 34, 36, 38, 40, 42, 44, 46, 49, 51, 53, 55, 57, 59, 61, 63, 66, 68, 70, 72, 74, 76, 78, 81, 83, 85, 87, 89, 91, 93, 95, 98, 100, 102, 104, 106, 108, 110, 113, 115, 117, 119, 121, 123, 125, 127, 130, 132

Offset

1,1

Comments

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.

Links

Table of n, a(n) for n=1..62.

Eric Weisstein's World of Mathematics, Beatty Sequence.

Index entries for sequences related to Beatty sequences

Formula

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.

Mathematica

Solve[1/Sinh[x] + 1/Cosh[x] == 1, x]
r = ArcSech[1/8 (4 - 4 Sqrt[2] - 9 Sqrt[5 + 4 Sqrt[2]] + (5 + 4 Sqrt[2])^(3/2))];
u = N[r, 250]
v = RealDigits[u][[1]];
Table[Floor[n*Sinh[r]], {n, 1, 150}]  (* A329938 *)
Table[Floor[n*Cosh[r]], {n, 1, 150}]  (* A329939 *)

Crossrefs

Cf. A329825, A329832, A329937, A329938 (complement).

Sequence in context: A187233 A247430 A329832 * A063459 A186329 A062417

Adjacent sequences: A329936 A329937 A329938 * A329940 A329941 A329942

Keyword

nonn,easy

Author

Clark Kimberling, Jan 02 2020

Status

approved