A334364 -id:A334364 - cp's OEIS frontend

A337729 a(n) = (4n+2)! Sum_{k=0..n} 1 / (4*k+2)!.

1, 361, 1819441, 43710250585, 3210080802962401, 563561785768079119561, 202205968733586788098486801, 132994909755454702268136738753721, 148026526435655214537290625514621562305, 262237873172349351865682580536682974917045801, 704454843460345510903820429747302209179158476142321

Offset: 0

Ilya Gutkovskiy, Sep 17 2020

nonn

Cf. A000522, A051396, A051397, A087350, A330045, A334364, A337725, A337726, A337727, A337728, A337730.

Table[(4 n + 2)! Sum[1/(4 k + 2)!, {k, 0, n}], {n, 0, 10}]
Table[(4 n + 2)! SeriesCoefficient[(1/2) (Cosh[x] - Cos[x])/(1 - x^4), {x, 0, 4 n + 2}], {n, 0, 10}]
Table[Floor[(1/2) (Cosh[1] - Cos[1]) (4 n + 2)!], {n, 0, 10}]

a(n) = (4*n+2)!*sum(k=0, n, 1/(4*k+2)!); \\ Michel Marcus, Sep 17 2020

E.g.f.: (1/2) * (cosh(x) - cos(x)) / (1 - x^4) = x^2/2! + 361*x^6/6! + 1819441*x^10/10! + 43710250585*x^14/14! + ...

a(n) = floor(c * (4*n+2)!), where c = (cosh(1) - cos(1)) / 2 = A334364.

A334363 Decimal expansion of Sum_{k>=0} 1/(4*k+1)!.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 24 2020

			1/1! + 1/5! + 1/9! + ... = 1.008336089225848981767442...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 36.

Cf. A001113, A049469, A073742, A143820, A332890, A334364, A334365.

RealDigits[(Sin[1] + Sinh[1])/2, 10, 110] [[1]]

suminf(k=0, 1/(4*k+1)!) \\ Michel Marcus, Apr 25 2020

Equals (sin(1) + sinh(1))/2.

A334365 Decimal expansion of Sum_{k>=0} 1/(4*k+3)!.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 24 2020

			1/3! + 1/7! + 1/11! + ... = 0.1668651044179524751...

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 209.
Index entries for transcendental numbers.

Cf. A001113, A049469, A073742, A332890, A334363, A334364.

RealDigits[(Sinh[1] - Sin[1])/2, 10, 111] [[1]]

suminf(k=0, 1/(4*k+3)!) \\ Michel Marcus, Apr 25 2020

Equals (sinh(1) - sin(1))/2.

A349804 Decimal expansion of cosh(1) - cos(1).

Original entry on oeis.org

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

Offset: 1

Christoph B. Kassir, Dec 11 2021

			1.00277832894710406107696901331408507...

Michael Ian Shamos, Shamos's catalog of the real numbers, (2011).

Cf. A073743, A049470, A334364, A334365.

RealDigits[Cosh[1] - Cos[1], 10, 100][[1]] (* Amiram Eldar, Dec 11 2021 *)

PARI
```
cosh(1) - cos(1)
```

Equals 2 * Sum_{k>=0} 1/(4*k+2)! = 2 * A334364. - Amiram Eldar, Dec 11 2021

cp's OEIS Frontend

A337729 a(n) = (4*n+2)! * Sum_{k=0..n} 1 / (4*k+2)!.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A334363 Decimal expansion of Sum_{k>=0} 1/(4*k+1)!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A334365 Decimal expansion of Sum_{k>=0} 1/(4*k+3)!.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A349804 Decimal expansion of cosh(1) - cos(1).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A337729 a(n) = (4n+2)! Sum_{k=0..n} 1 / (4*k+2)!.