A344505 -id:A344505 - cp's OEIS frontend

A175316 Decimal expansion of coth(Pi).

1, 0, 0, 3, 7, 4, 1, 8, 7, 3, 1, 9, 7, 3, 2, 1, 2, 8, 8, 2, 0, 1, 5, 5, 2, 6, 9, 1, 1, 9, 4, 8, 0, 0, 0, 1, 7, 4, 6, 2, 4, 5, 2, 4, 2, 2, 9, 9, 5, 9, 0, 7, 6, 6, 3, 4, 0, 4, 8, 3, 0, 0, 2, 8, 4, 6, 7, 0, 0, 4, 6, 7, 6, 3, 3, 0, 5, 8, 0, 7, 1, 9, 6, 8, 8, 0, 5, 8, 5, 6, 2, 2, 6, 6, 1, 1, 1, 8, 1, 1, 0, 8, 2, 0, 6, 1, 2, 8, 4

Offset: 1

R. J. Mathar, Apr 01 2010

			1.003741873197321288201552691194800017462452422...

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 28.

Cf. A175314, A175315, A019692, A344505.

Tau := 2*Pi: Digits := 120: (exp(Tau) + 1)/(exp(Tau) - 1):
evalf(%)*10^110: ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, May 21 2021

RealDigits[Coth[Pi], 10, 110][[1]] (* Georg Fischer, Apr 03 2020 *)

Equals A175314 divided by A175315.
Equals (exp(tau) + 1)/(exp(tau) - 1), where tau = 2*Pi = A019692. - Peter Luschny, May 22 2021
From Wolfe Padawer, Jan 28 2023: (Start)
Equals Sum_{k>=-oo} 1/(Pi + Pi*k^2).
Equals 1 + Sum_{k>=0} 2*e^(-2*Pi*k - 2*Pi).
Equals 1 + lim_{n->oo} (2*e^(-2*Pi*n))*(e^(2*Pi*n) - 1)/(e^(2*Pi) - 1). (End)
Equals 1/A344505. - Hugo Pfoertner, Jun 24 2025

a(42) ff. corrected by Georg Fischer, Apr 03 2020

A372822 a(n) = floor(n*tanh(Pi/2)).

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 61, 62, 63

Offset: 0

Paolo Xausa, Jul 04 2024

Paolo Xausa, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Beatty Sequence.
Index entries for sequences related to Beatty sequences

Cf. A000796, A344505, A367960, A374323.

Mathematica
```
Floor[Range[0, 100]*Tanh[Pi/2]]
```

A096613 Values of k such that floor(k*tanh(Pi)) = floor((k+1) tanh(Pi)).

Original entry on oeis.org

268, 536, 804, 1072, 1341, 1609, 1877, 2145, 2414, 2682, 2950, 3218, 3487, 3755, 4023, 4291, 4560, 4828, 5096, 5364, 5633, 5901, 6169, 6437, 6706, 6974, 7242, 7510, 7779, 8047, 8315, 8583, 8852, 9120, 9388, 9656, 9925, 10193, 10461, 10729

Offset: 1

Eric W. Weisstein, Jun 30 2004

			Floor(k*tanh(Pi)) = k-1 for 1<=k<=267, but floor(268*tanh(Pi)) = floor(269*tanh(Pi)) = 267, so 268 is the first member of the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Almost Integer.

Cf. A344505.

Select[Range[10^4], Floor[#*Tanh[Pi]] == Floor[(# + 1)*Tanh[Pi]] &] (* Amiram Eldar, Apr 14 2022 *)
With[{c=Tanh[Pi]},SequencePosition[Table[Floor[c* n],{n,11000}],{x_,x_}][[;;,1]]] (* Harvey P. Dale, Aug 29 2025 *)

A347058 Decimal expansion of (1 + tanh(Pi)) / 2.

Original entry on oeis.org

9, 9, 8, 1, 3, 6, 0, 3, 8, 1, 1, 0, 3, 7, 4, 9, 7, 2, 1, 3, 2, 3, 4, 5, 2, 9, 0, 0, 0, 6, 2, 6, 8, 3, 5, 5, 9, 4, 8, 4, 4, 9, 5, 9, 5, 4, 0, 2, 2, 9, 3, 8, 0, 7, 1, 8, 1, 3, 0, 6, 2, 0, 7, 9, 8, 9, 2, 7, 0, 6, 4, 9, 4, 9, 1, 4, 1, 0, 8, 5, 8, 8, 4, 2, 6, 4, 1

Offset: 0

Sean A. Irvine, Aug 14 2021

			0.99813603811037497213234529000626835594844959540...

L. B. W. Jolley, Summation of Series, Dover, 1961, Eq. (944).
Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A216707, A344505.

RealDigits[(1 + Tanh[Pi])/2, 10, 120][[1]] (* Amiram Eldar, Jun 07 2023 *)

Equals exp(2*Pi) / (1 + exp(2*Pi)).

Equals Sum_{k>=0} (-1)^k * exp(-2*Pi*k).

cp's OEIS Frontend

A175316 Decimal expansion of coth(Pi).

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A372822 a(n) = floor(n*tanh(Pi/2)).

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A096613 Values of k such that floor(k*tanh(Pi)) = floor((k+1) tanh(Pi)).

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A347058 Decimal expansion of (1 + tanh(Pi)) / 2.

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