A309419 -id:A309419 - cp's OEIS frontend

A309091 Decimal expansion of 4/(Pi-2).

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

Offset: 1

Alois P. Heinz, Jul 11 2019

This can be computed using a recursion formula discovered by an algorithm called "The Ramanujan Machine":
                             1*3
    4/(Pi-2)  =  3 + --------------------
                               2*4
                     5 + ----------------
                                 3*5
                         7 + ------------
                                   4*6
                             9 + --------
                                 11 + ...  .
  For a proof by humans see the arXiv:1907.00205 preprint linked below.

			3.50387678776821732240781940302290775850079601361148312728094190...

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Gal Raayoni, George Pisha, Yahel Manor, Uri Mendlovic, Doron Haviv, Yaron Hadad, and Ido Kaminer, The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants, arXiv:1907.00205 [cs.LG], 2019-2020.
The Ramanujan Machine, Using algorithms to discover new mathematics.
Index entries for transcendental numbers.

Cf. A000796, A005563, A144396, A309419, A309420.

nn:= 126: # number of digits
b:= i-> `if`(i<2*nn, 2*i+1 +i*(i+2)/b(i+1), 1):
evalf(b(1), nn);

RealDigits[4/(Pi-2), 10, 120][[1]] (* Amiram Eldar, Jun 29 2023 *)

A309420 Decimal expansion of 4/(3*Pi-8).

Original entry on oeis.org

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

Offset: 1

Alois P. Heinz, Jul 30 2019

Conjecturally, this can be computed using a recursion formula discovered by an algorithm called "The Ramanujan Machine":
                                1*1
    4/(3*Pi-8)  =  3 - --------------------
                                  2*3
                       6 - ----------------
                                    3*5
                           9 - ------------
                                      4*7
                               12 - --------
                                    15 - ...  .

			2.80745499308537947657159669392697176828889127774792...

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Gal Raayoni, George Pisha, Yahel Manor, Uri Mendlovic, Doron Haviv, Yaron Hadad, Ido Kaminer, The Ramanujan Machine: Automatically Generated Conjectures on Fundamental Constants, arXiv:1907.00205 [cs.LG], 2019-2020.
The Ramanujan Machine, Using algorithms to discover new mathematics
Index entries for transcendental numbers

Cf. A000796, A309091, A309419.

nn:= 126: # number of digits
# b:= i-> `if`(i<4*nn, 3*i -i*(2*i-1)/b(i+1), 1):
# evalf(b(1), nn);
evalf(4/(3*Pi-8), nn);

RealDigits[4/(3 Pi-8),10,120][[1]] (* Harvey P. Dale, May 09 2021 *)

A334397 Decimal expansion of (e - 2)/e.

Original entry on oeis.org

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

Offset: 0

Daniel Hoyt, Apr 26 2020

			0.2642411176571153568089524596770782651...

M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.

Cf. A001008, A001113, A002805, A019739, A135002, A309419, A334396.

RealDigits[1 - 2/E, 10, 100][[1]] (* Alonso del Arte, Apr 26 2020 *)

1 - 2/exp(1) \\ Michel Marcus, May 01 2020

Equals Integral_{x=0..1} x/e^x dx.
Equals 1 - A135002.
Equals 1/A309419.
Equals -Integral_{x=0..1, y=0..1} x*y/(exp(x*y)*log(x*y)) dx dy. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to the above integral.) - Petros Hadjicostas, Jun 30 2020
From Amiram Eldar, Aug 05 2020: (Start)
Equals Sum_{k>=0} (-1)^k/(k! * (k+2)).
Equals Sum_{k>=1} 1/((2*k)! * (k+1)).
Equals Sum_{k>=1} (-1)^k * k^2 * H(k)/k!, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number. (End)

A368270 a(n) = Sum_{k=0..n} 2^(n-k) * k^n.

Original entry on oeis.org

1, 1, 6, 47, 490, 6417, 101178, 1866139, 39425322, 938856053, 24883226698, 726510389607, 23169961642698, 801435579830329, 29884247978965146, 1195036047465095027, 51016725208899539626, 2315820594694418639325, 111384453953719146198762

Offset: 0

Seiichi Manyama, Dec 25 2023

Cf. A031971, A120485, A349963.

PARI
```
a(n) = sum(k=0, n, 2^(n-k)*k^n);
```

a(n) ~ A309419 * n^n. - Vaclav Kotesovec, Dec 26 2023

cp's OEIS Frontend

A309091 Decimal expansion of 4/(Pi-2).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A309420 Decimal expansion of 4/(3*Pi-8).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A334397 Decimal expansion of (e - 2)/e.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A368270 a(n) = Sum_{k=0..n} 2^(n-k) * k^n.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula