A334401 -id:A334401 - cp's OEIS frontend

A334402 Decimal expansion of cosh(Pi).

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

Offset: 2

Ilya Gutkovskiy, Apr 26 2020

This constant is transcendental.

			(e^Pi + e^(-Pi))/2 = 11.5919532755215206277517520525601...

Index entries for transcendental numbers

Cf. A000796, A001113, A039661, A068470, A073743, A093580, A175314, A308715, A323982, A334401.

Mathematica
```
RealDigits[Cosh[Pi], 10, 110] [[1]]
```

Equals Sum_{k>=0} Pi^(2*k)/(2*k)!.
Equals Product_{k>=0} (1 + 4/(2*k+1)^2).
Equals Product_{k>=1} (k^2 + 4)/(k^2 + 1). - Amiram Eldar, Aug 09 2020

A330864 Decimal expansion of sinh(Pi/2)/2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 28 2020

This constant is transcendental.

			(1 + 1/2^2) * (1 - 1/3^2) * (1 + 1/4^2) * (1 - 1/5^2) * (1 + 1/6^2) * ... = (e^(Pi/2) - e^(-Pi/2))/4 = 1.15064945115364743673152001...

Index entries for transcendental numbers

Cf. A042972, A049006, A090986, A156648, A175615, A308715, A308716, A308718, A323983, A330865, A334401.

Mathematica
```
RealDigits[Sinh[Pi/2]/2, 10, 110] [[1]]
```

sinh(Pi/2)/2 \\ Michel Marcus, Apr 28 2020

Equals Sum_{k>=1} Pi^(2*k-1)/(4^k*(2*k-1)!).
Equals Product_{k>=2} (1 + (-1)^k/k^2).
Equals (i^(-i) - i^i)/4, where i is the imaginary unit.

A356479 Decimal expansion of (sqrt(3)/Pi) * sinh(Pi/sqrt(3)).

Original entry on oeis.org

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

Offset: 1

Christoph B. Kassir, Aug 08 2022

			1.6459025152253961193544118815663276416192231065463...

Michael Ian Shamos, Shamos's catalog of the real numbers, (2011) [page 520, but misprint in the product 3*k should be 3*k^2].

Cf. A000796, A002194, A334401.

RealDigits[Sqrt[3]*Sinh[Pi/Sqrt[3]]/Pi, 10, 100][[1]] (* Amiram Eldar, Aug 09 2022 *)

PARI
```
print((sqrt(3)/Pi) * sinh(Pi/sqrt(3)))
```

Equals Product_{k>=1} (1 + 1/(3*k^2)).

A361260 Maximum latitude in degrees of spherical Mercator projection with an aspect ratio of one, arctan(sinh(Pi))*180/Pi.

Original entry on oeis.org

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

Offset: 2

Donghwi Park, Mar 06 2023

Widely used as a cutoff line of web maps which use the web Mercator projection.

			85.05112877980659237779671552192469206698259126842068...

Cf. A334401.

RealDigits[ArcTan[Sinh[Pi]]/Degree, 10, 100][[1]] (* Amiram Eldar, Mar 06 2023 *)

atan(sinh(Pi))*180/Pi \\ Michel Marcus, Mar 06 2023

Equals arctan(sinh(Pi))*180/Pi.

Equals 360/Pi*arctan(exp(Pi)) - 90.

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A334402 Decimal expansion of cosh(Pi).

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A330864 Decimal expansion of sinh(Pi/2)/2.

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A356479 Decimal expansion of (sqrt(3)/Pi) * sinh(Pi/sqrt(3)).

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A361260 Maximum latitude in degrees of spherical Mercator projection with an aspect ratio of one, arctan(sinh(Pi))*180/Pi.

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Formula