A268683 -id:A268683 - cp's OEIS frontend

A268682 Decimal expansion of 1 - 1/sqrt(2).

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

Offset: 0

Charles R Greathouse IV, Feb 19 2016

This is the maximum fraction of mass-energy of a black hole which can come from angular momentum, and hence the maximum energy which can be extracted from the black hole via the Penrose process.
Differs from A157215 only in one or two leading digits. - R. J. Mathar, Feb 24 2016
This is the probability that a randomly selected vertex in a random Schroeder tree is a leaf as the number of leaves goes to infinity. See Corollary 2.1.2. of Van Duzer. - Michel Marcus, Apr 12 2019

			0.29289321881345247559915563789515096071516406231152596341166013100463376...

Charles D. Dermer and Govind Menon, High Energy Radiation from Black Holes: Gamma Rays, Cosmic Rays, and Neutrinos (2009). See pp. 400-402.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Anthony Van Duzer, Subtrees of a Given size in Schroeder Trees, arXiv:1904.05525 [math.CO], 2019.
Stijn J. van Tongeren, Rotating black holes (2009).
Wikipedia, Penrose process
Index entries for algebraic numbers, degree 2

Cf. A010503, A268683.

1-1/Sqrt(2); // Vincenzo Librandi, Feb 20 2016

RealDigits[1 - 1 / Sqrt[2], 10, 90] [[1]] (* Vincenzo Librandi, Feb 20 2016 *)

PARI
```
1-sqrt(.5)
```

Equals 1 - A010503.
a(n) = 9 - A010503(n). - Philippe Deléham, Feb 21 2016
Equals Integral_{x=0..Pi/4} sin(x) dx. - Amiram Eldar, Jun 29 2020

More digits from Jon E. Schoenfield, Mar 15 2018

A193355 Decimal expansion of Pi/(2 + 2*sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Frank M Jackson, Jul 24 2011

This is the first of the three angles (in radians) of a unique triangle that is right angled and where the angles are in a harmonic progression: Pi/(2+2*sqrt(2)) (this sequence), Pi/(2+sqrt(2)) (A193373), Pi/2 (A019669). The angles (in degrees) are approximately 37.279, 52.721, 90. The common difference between the denominators of the harmonic progression is sqrt(2).

			0.6506451422...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Index entries for transcendental numbers

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)/(2 + 2*Sqrt(2)); // G. C. Greubel, Sep 29 2018

evalf(Pi/(2+2*sqrt(2)),120); # Muniru A Asiru, Sep 30 2018

N[Pi/(2 + 2*Sqrt[2]), 100]
RealDigits[Pi/(2 + 2*Sqrt[2]), 10, 100][[1]] (* G. C. Greubel, Sep 29 2018 *)

default(realprecision,100); Pi/(2+2*sqrt(2))

Equals Pi/(2+2*sqrt(2)).
Equals Integral_{x=0..Pi/2} cos(x)^2/(1 + sin(x)^2) dx = Integral_{x=0..Pi/2} sin(x)^2/(1 + cos(x)^2) dx. - Amiram Eldar, Aug 16 2020
Equals 4*Sum_{k >= 0} (-1)^k/((4*k + 1)*(4*k + 2)*(4*k + 3)). - Peter Bala, Jul 15 2024
Equals Integral_{x=0..1} sqrt(1 - x^2)/(1 + x^2) dx. - Kritsada Moomuang, Jun 05 2025
Equals A247719 - A019669 = A000796*A268683. - R. J. Mathar, Jul 22 2025

A319905 Decimal expansion of 4*(sqrt(2) - 1)/3.

Original entry on oeis.org

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

Offset: 0

Franck Maminirina Ramaharo, Oct 01 2018

A 90-degree unit-circular arc in the first quadrant can be approximated by a cubic Bézier curve. In this case, L = 4*(sqrt(2) - 1)/3 is the unit tangent vector scaling factor that minimizes the distance between the curve and the unit circle segment, provided its endpoints and midpoint are interpolated.
Riškus referred to this constant as "magic number".
The Bézier curve with control points {(1,0), (1,L), (L,1), (0,1)} has a minimum distance to the origin of 1 (at t in {0, 1/2, 1}), and it has a maximum distance to the origin of (1/3)*sqrt(71/6-2*sqrt(2)) = 1.00027253... at t in {(3 - sqrt(3))/6,(3 + sqrt(3))/6}. - Peter Kagey, Feb 21 2025

			0.552284749830793398402251632279597438092895833835930...

Tor Dokken, Morten Dæhlen, Tom Lyche and Knut Mørken, Good approximation of circles by curvature-continuous Bézier curves, Computer Aided Geometric Design Vol. 7 (1990), 33-41.
Aleksas Riškus, Approximation of a cubic Bézier curve by circular arcs and vice versa, Information Technology And Control Vol. 35 (2006), 371-378.
Adam G. Stanislav, Drawing a circle with Bézier Curves
Wikipedia, Bézier curve
Wikipedia, Composite Bézier curve

Cf. A002193, A156309, A188582, A268683.

Maple
```
Digits:=1000; evalf(4*(sqrt(2) - 1)/3);
```

RealDigits[4*(Sqrt[2] - 1)/3, 10, 100][[1]]

PARI
```
4*(sqrt(2) - 1)/3
```

Equals (4/3)*tan(Pi/8).

Irrational number represented by the periodic continued fraction [0; [1, 1, 4, 3]]; positive real root of 9*x^2 + 24*x - 16. - Peter Luschny, Oct 04 2018

A351898 Decimal expansion of metallic ratio for N = 14.

Original entry on oeis.org

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

Offset: 2

A.H.M. Smeets, Feb 24 2022

Decimal expansion of continued fraction [14; 14, 14, 14, ...].
Also largest solution of x^2 - 14 x - 1 = 0.
Essentially the same digit sequence as A010503, A157214, A174968 and A268683.
The metallic ratio's for N = A077444(n) are equal to powers of the silver ratio, i.e., A014166^(2n-1); this constant represents the special case for N = A077444(2).

			14.0710678118654752440084436210484903928483593...

Michael Penn, 7+5√2 is a perfect cube??, YouTube video, 2022.
Wikipedia, Metallic mean.

Cf. A010503, A077444, A157214, A174968, A268683.

Metallic ratios: A001622 (N=1), A014176 (N=2), A098316 (N=3), A098317 (N=4), A098318 (N=5), A176398 (N=6), A176439 (N=7), A176458 (N=8), A176522 (N=9), A176537 (N=10), A244593 (N=11).

RealDigits[7 + 5*Sqrt[2], 10, 100][[1]] (* Amiram Eldar, Feb 24 2022 *)

PARI
```
(1+sqrt(2))^3
```

Equals 2 + 5*A014176.
Equals A014176^3.
Equals exp(arcsinh(7)). - Amiram Eldar, Jul 04 2023

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A268682 Decimal expansion of 1 - 1/sqrt(2).

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A193355 Decimal expansion of Pi/(2 + 2*sqrt(2)).

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A319905 Decimal expansion of 4*(sqrt(2) - 1)/3.

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A351898 Decimal expansion of metallic ratio for N = 14.

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