A309091 -id:A309091 - cp's OEIS frontend

A309419 Decimal expansion of e/(e-2).

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

Offset: 1

Alois P. Heinz, Jul 30 2019

This can be computed using a recursion formula discovered by an algorithm called "The Ramanujan Machine":
                             1
    e/(e-2)  =  4 - --------------------
                               2
                    5 - ----------------
                                 3
                        6 - ------------
                                   4
                            7 - --------
                                 8 - ...  .
  For a proof by humans see the arXiv:1907.00205 preprint linked below.

			3.78442238235466562875310575695963305674795677063...

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.

Cf. A001113, A309091, A309420, A334397.

nn:= 126: # number of digits
b:= i-> `if`(i

RealDigits[E/(E-2), 10, 120][[1]] (* Amiram Eldar, Jun 28 2023 *)

Equals 1/A334397.

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 *)

A330156 Decimal expansion of the continued fraction expansion [1; 1/2, 1/3, 1/4, 1/5, 1/6, ...].

Original entry on oeis.org

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

Offset: 1

Daniel Hoyt, Dec 03 2019

This constant is formed from the continued fraction [1; 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ...] the reciprocals of the positive integers, A000027.

			1.7519383938841086612039097015114538792503980068057415636404709501399828870437...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4.4, p. 23.

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 562.
Wikipedia, Continued Fraction Expansions of Pi.
Index entries for transcendental numbers.

Cf. A000027, A000796, A052119, A073824, A309091.

First[RealDigits[2/(Pi - 2), 10, 100]] (* Paolo Xausa, Apr 27 2024 *)

2 / (Pi - 2) \\ Michel Marcus, Dec 05 2019

1/atan(cotan(1)) \\ Daniel Hoyt, Apr 11 2020

Equals 2 / (Pi - 2).
Equals 1/arctan(cot(1)). - Daniel Hoyt, Apr 11 2020
From Stefano Spezia, Oct 26 2024: (Start)
2/(Pi - 2) = 1 + K_{n>=1} n*(n+1)/1, where K is the Gauss notation for an infinite continued fraction. In the expanded form, 2/(Pi - 2) = 1 + 1*2/(1 + 2*3/(1 + 3*4/(1 + 4*5/(1 + 5*6/(1 + ...))))) (see Finch at p. 23).
2/(Pi - 2) = Sum_{n>=1} (2/Pi)^n (see Shamos). (End)
Equals A309091/2. - Hugo Pfoertner, Oct 28 2024

A342977 Decimal expansion of (Pi - 2) / 4.

Original entry on oeis.org

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

Offset: 0

Michal Paulovic, Apr 01 2021

The constant represents the area of a circular segment bounded by an arc of Pi/2 radians (the right angle) of a unit circle and by a chord of the length of sqrt(2). Four such segments result when a square with the side length of sqrt(2) is circumscribed by a unit circle. The area of each segment is:
  A = (R^2 / 2) * (theta - sin(theta))
  A = (1^2 / 2) * (Pi/2 - sin(Pi/2))
  A = (1 / 2) * (Pi/2 - 1)
  A = (Pi - 2) / 4 = 0.28539816...
  where Pi = 3.14159265... (A000796) is the area bounded by the unit circle, and 2 is the area of the inscribed square.
Apart from the first digit this is the same as Pi/4 = 0.78539816... (A003881), the area of a circular sector bounded by the arc of Pi/2 = 1.57079632... (A019669) radians of the unit circle and by two radii of unit length, and 1/2 = 0.5 (A020761) is one-quarter of the area of the inscribed square.
The constant is close to 2/7 = 0.28571428... (2 * A020806) and Pi/11 = 0.28559933... (A019678). The equation (x - 2)/4 = x/11 has a solution x = 22/7 = 3.14285714... (A068028), which is an approximation of Pi.
The best rational approximation of the constant using small positive integers (less than 1000) is 129/452 = 0.28539823..., the next best approximation is 4771/16717 = 0.28539809...
The reciprocal of the constant is:
  1/A = 4 / (Pi - 2) = 3.50387678... (A309091).
The sagitta (height) of the circular segment is:
  h = R * (1 - cos(theta/2))
  h = 1 * (1 - cos(Pi/4))
  h = 1 - sqrt(2) / 2
  h = 1 - 1 / sqrt(2) = 0.29289321... (A268682).

			0.2853981633974483...

Index entries for transcendental numbers

Cf. A000796, A019669, A019678, A020761, A020806, A068028, A268682, A309091. Essentially the same as A003881.

RealDigits[Pi/4 - 1/2, 10, 100][[1]] (* Amiram Eldar, Jun 08 2021 *)

PARI
```
(Pi - 2) / 4
```

Equals Integral_{x=-sqrt(2)/2..sqrt(2)/2} Integral_{y=sqrt(2)/2..sqrt(1-x^2)} dy dx.
Equals Sum_{k>=1} (-1)^(k + 1)/(4*k^2 - 1). - Amiram Eldar, Jun 08 2021
Continued fraction: 1/(3 + 3/(4 + 15/(4 + 35/(4 + ... + (4*n^2 - 1)/(4 + ...). - Peter Bala, Feb 22 2024

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A309419 Decimal expansion of e/(e-2).

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

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A330156 Decimal expansion of the continued fraction expansion [1; 1/2, 1/3, 1/4, 1/5, 1/6, ...].

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A342977 Decimal expansion of (Pi - 2) / 4.

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