cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 16, 18, 20, 22, 24, 26, 28, 30, 32, 33, 35, 37, 39, 41, 43, 45, 47, 48, 50, 52, 54, 56, 58, 60, 62, 64, 65, 67, 69, 71, 73, 75, 77, 79, 80, 82, 84, 86, 88, 90, 92, 94, 96, 97, 99, 101, 103, 105, 107, 109, 111, 112, 114, 116, 118
Offset: 1

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Author

Clark Kimberling, Jan 02 2020

Keywords

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

Crossrefs

Cf. A329825, A329831, A329937, A329939 (complement).

Programs

  • 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 *)

Formula

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.

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

Original entry on oeis.org

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

Views

Author

Clark Kimberling, Jan 02 2020

Keywords

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

Crossrefs

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

Programs

  • 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 *)

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.
Showing 1-2 of 2 results.

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