A323982 -id:A323982 - cp's OEIS frontend

A323983 Decimal expansion of sinh(Pi/6).

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

Offset: 0

N. J. A. Sloane, Feb 11 2019

Decimal expansion of sum sigma_2 involved in calculation of i^i. [Uhler]

			0.5478534738880398084757156477174314033527835217758617619908956382877300771661...

Robert G. Wilson v, Table of n, a(n) for n = 0..1000
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.

Cf. A323982, A323984, A323985, A323986, A323987.

a[0] = a[1] = 1; a[n_] := ((n - 2)^2 + 1) a[n - 2]; t[x_] := N[(1/2)^x*a[x]/x!, 128]; RealDigits[Sum[t[x], {x, 1, 421, 2}], 10, 111][[1]] (* Robert G. Wilson v, Feb 16 2019 *)
RealDigits[Sinh[Pi/6], 10, 120][[1]] (* Vaclav Kotesovec, Feb 16 2019 *)

A323982 + A323983 = A323984.

More digits from Robert G. Wilson v, Feb 16 2019

Simpler definition from Vaclav Kotesovec, Feb 16 2019

A323984 Decimal expansion of exp(Pi/6).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 11 2019

			1.6880917949644686006168476280967822941196811892871823125684056078889433931696...

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
Index entries for transcendental numbers.

Cf. A323982, A323983, A323985, A323986, A323987.

RealDigits[Exp[Pi/6], 10, 111][[1]] (* or *)
a[0] = a[1] = 1; a[n_] := ((n - 2)^2 + 1) a[n - 2]; t[x_] := N[(1/2)^x*a[x]/x!, 128]; RealDigits[Sum[t[x], {x, 0, 421}], 10, 111][[1]] (* Robert G. Wilson v, Feb 16 2019 *)

A323982 + A323983 = A323984.

Corrected and extended by Robert G. Wilson v, Feb 16 2019

A323985 Decimal expansion of exp(-Pi/6).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 11 2019

			0.5923848471883889836654163326619194874141141457354587885866143313134832388373...

Robert G. Wilson v, Table of n, a(n) for n = 0..1000
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
Index entries for transcendental numbers.

Cf. A323982, A323983, A323984, A323986, A323987.

RealDigits[Exp[-Pi/6], 10, 111][[1]] (* Robert G. Wilson v, Feb 16 2019 *)

Corrected and extended by Robert G. Wilson v, Feb 16 2019

A323986 Decimal expansion of exp(Pi/3).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Feb 11 2019

			2.8496539082263614974741273198529043939640061027811268817432387784881424491588...

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
Index entries for transcendental numbers.

Cf. A323982, A323983, A323984, A323985, A323987.

RealDigits[Exp[Pi/3], 10, 111][[1]] (* Robert G. Wilson v, Feb 16 2019 *)

Corrected and extended by Robert G. Wilson v, Feb 16 2019

A323987 Decimal expansion of exp(-Pi/3).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Feb 11 2019

			0.3509198071784109675657367159969530583625731536209617406523848978345471913879...

Robert G. Wilson v, Table of n, a(n) for n = 0..1000
H. S. Uhler, On the numerical value of i^i, Amer. Math. Monthly, 28 (1921), 114-116.
Index entries for transcendental numbers.

Cf. A323982, A323983, A323984, A323985, A323986.

RealDigits[Exp[-Pi/3], 10, 111][[1]] (* Robert G. Wilson v, Feb 16 2019 *)

Corrected and extended by Robert G. Wilson v, Feb 16 2019

A334402 Decimal expansion of cosh(Pi).

Original entry on oeis.org

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

A330865 Decimal expansion of cosh(Pi/2)/Pi.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 28 2020

			(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))/(2*Pi) = 0.7986963159564630848638067...

Cf. A042972, A049006, A049541, A156648, A175615, A308715, A308716, A308718, A323982, A330864, A334402.

RealDigits[Cosh[Pi/2]/Pi, 10, 110] [[1]]

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

Equals Sum_{k>=0} Pi^(2*k-1)/(4^k*(2*k)!).
Equals Product_{k>=2} (1 - (-1)^k/k^2).
Equals (i^(-i) + i^i)/(2*Pi), where i is the imaginary unit.

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A323983 Decimal expansion of sinh(Pi/6).

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A323984 Decimal expansion of exp(Pi/6).

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A323985 Decimal expansion of exp(-Pi/6).

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A323986 Decimal expansion of exp(Pi/3).

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A323987 Decimal expansion of exp(-Pi/3).

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A334402 Decimal expansion of cosh(Pi).

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A330865 Decimal expansion of cosh(Pi/2)/Pi.

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