A180434 -id:A180434 - cp's OEIS frontend

A019692 Decimal expansion of 2*Pi.

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

Offset: 1

N. J. A. Sloane

Pi/5 or 2*Pi/10 is the expected surface area containing completely a Brownian curve (trajectory) on a plane. - Lekraj Beedassy, Jul 28 2005
Bob Palais considers this a more fundamental constant than Pi, see the Palais reference and link. - Jonathan Vos Post, Sep 10 2010
The Persian mathematician Jamshid al-Kashi seems to have been the first to use the circumference divided by the radius as the circle constant. In Treatise on the Circumference published 1424 he calculated the circumference of a unit circle to 9 sexagesimal places. - Peter Harremoës, John W. Nicholson, Aug 02 2012
"Proponents of a new mathematical constant tau (τ), equal to two times π, have argued that a constant based on the ratio of a circle's circumference to its radius rather than to its diameter would be more natural and would simplify many formulas" (from Wikipedia). - Jonathan Sondow, Aug 15 2012
The constant 2*Pi appears in the formula for the period T of a simple gravity pendulum. For small angles this period is given by Christiaan Huygens’s law, i.e., T = 2*Pi*sqrt(L/g), see for more information A223067. - Johannes W. Meijer, Mar 14 2013
There are seven consecutive nines at positions 762 to 768. - Roland Kneer, Jul 05 2013
Volume of a cylinder in which a sphere of radius 1 can be inscribed. - Omar E. Pol, Sep 25 2013
2*Pi is also the surface area of a sphere whose diameter equals the square root of 2. More generally, x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 18 2013
From Bernard Schott, Jan 31 2020: (Start)
Also, (2*Pi)*a^2 is the area of the deltoid (an hypocycloid with three cusps) whose Cartesian parametrization is:
   x = a * (2*cos(t) + cos(2*t)),
   y = a * (2*sin(t) - sin(2*t)).
The length of this deltoid is 16*a. See the curve at the Mathcurve link. (End)
Pi/5 = 0.1 * 2*Pi is the mean area of the plane triangles formed by 3 points independently and uniformly chosen at random on the surface of a unit-radius sphere. - Amiram Eldar, Aug 06 2020

			6.283185307179586476925286766559005768394338798750211641949889184615632...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4, p. 17.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 69.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Robert Ferréol, Deltoid, Mathcurve.
Christophe Garban and José A. Trujillo Ferreras, The expected area of the filled planar Brownian loop is pi/5, Communications in mathematical physics, Vol. 264, No. 3 (2006), pp. 797-810, preprint, arXiv:math/0504496 [math.PR], 2005.
Peter Harremoës, Al-Kashi’s constant
Michael Hartl, The Tau Manifesto.
Melissa Larson, Verifying and discovering BBP-type formulas, 2008.
Bob Palais, Web page about "Pi is wrong!".
Bob Palais, Pi is wrong!, The Mathematical Intelligencer Volume 23, Number 3, 2001, pp. 7-8.
Grant Sanderson, How pi was almost 6.283185..., 3Blue1Brown video (2018).
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 318.
Wikipedia, Tau proposals.
Wikipedia, Bailey-Borwein-Plouffe formula.
Index entries for transcendental numbers.

Cf. A058291 (continued fraction).
Cf. A000796, A019693, A019699, A091925, A244979.
Cf. A093828 (astroid), A180434 (loop of strophoid), A197723 (cardioid).

using Nemo
RR = RealField(334)
tau = const_pi(RR) + const_pi(RR)
tau |> println # Peter Luschny, Mar 14 2018

R:= RealField(100); 2*Pi(R); // G. C. Greubel, Mar 08 2018

RealDigits[N[2 Pi, 6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

default(realprecision, 20080); x=2*Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019692.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009

# Use some guard digits when computing.
# BBP formula P(1, 16, 8, (0, 8, 4, 4, 0, 0, -1, 0)).
from decimal import Decimal as dec, getcontext
def BBPtau(n: int) -> dec:
    getcontext().prec = n
    s = dec(0); f = dec(1); g = dec(16)
    for k in range(n):
        ek = dec(8 * k)
        s += f * ( dec(8) / (ek + 2) + dec(4) / (ek + 3)
                 + dec(4) / (ek + 4) - dec(1) / (ek + 7))
        f /= g
    return s
print(BBPtau(200))  # Peter Luschny, Nov 03 2023

e^(Zeta'(0)/Zeta(0)) = 2*Pi. - Peter Luschny, Jun 17 2018
From Peter Bala, Oct 30 2019: (Start)
2*Pi = Sum_{n >= 0} (-1)^n*( 1/(n + 1/6) + 1/(n + 5/6) ).
2*Pi = Sum_{n >= 0} (-1)^n*( 1/(n + 1/10) - 1/(n + 3/10) - 1/(n + 7/10) + 1/(n + 9/10) ). Cf. A091925 and A244979. (End)
From Amiram Eldar, Aug 06 2020: (Start)
Equals Gamma(1/6)*Gamma(5/6).
Equals Integral_{x=0..oo} log(1 + 1/x^6) dx.
Equals Integral_{x=0..oo} log(1 + 4/x^2) dx.
Equals Integral_{x=-oo..oo} exp(x/6)/(exp(x) + 1) dx.
Equals Sum_{k>=0} 1/((k + 1/4)*(k + 3/4)). (End)
Equals 4*Integral_{x=0..1} 1/sqrt(1 - x^2) dx (see Finch). - Stefano Spezia, Oct 19 2024

A019694 Decimal expansion of 2*Pi/5.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Also, with proper offset, decimal expansion of the magnetic permeability of vacuum in SI units, mu_0 = 4*Pi*10^-7 N A^-2, an assigned metrological constant. [This exact expression for mu_0 was valid until the 2019 SI redefinition. In the New SI, mu_0 is numerically very close to that value but is determined only up to a certain error. - Andrey Zabolotskiy, May 22 2019]
Regarding these, see A003678 for general context notes, references and links. - Stanislav Sykora, Jun 16 2012
With offset 2 this is also the decimal expansion of 4*Pi, the surface area of a sphere whose diameter equals the square root of 4, hence its radius is 1. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Jan 18 2013, Oct 05 2013, Dec 22 2013
4*Pi is also the area of the domain bounded by the witch of Agnesi whose Cartesian equation is y = 8 / (x^2 + 4) and its asymptote. More generally (4*Pi) * a^2 is the area of the domain bounded by the witch of Agnesi whose Cartesian equation is y = (8*a^3) / (x^2 + 4*a^2) and its asymptote (Eric Weisstein's link, formula 6). - Bernard Schott, Jun 28 2023

			1.2566370614359172953850573533118....
mu_0 = 12.566370614359172953850573533118... 10^-7 N/A^2. - _Stanislav Sykora_, Jun 16 2012

Ivan Panchenko, Table of n, a(n) for n = 1..1000
NIST, magnetic constant mu_0.
Nobelprize.org, The Discovery of Quasicrystals (PDF).
Eric Weisstein's World of Mathematics, Witch of Agnesi.
Wikipedia, Crystallographic Restriction Theorem.
Index entries for sequences related to curves.
Index entries for transcendental numbers.

Other assigned constants: A003678, A072915, A081799, A182999, A213610, A213611, A213612, A213613, A213614.

Cf. A093828 (astroid), A180434 (loop of strophoid), A197723 (cardioid), A336266 (double egg), A336308 (ovoid).

Digits:=100: evalf(2*Pi/5); # Wesley Ivan Hurt, Jan 07 2017

RealDigits[ 2*Pi/5, 10, 111][[1]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 and modified by Robert G. Wilson v *)

PARI
```
2*Pi/5 \\ G. C. Greubel, Sep 11 2017
```

Equals Sum_{k>=1} sin(Pi*k/5)/k. - Amiram Eldar, Aug 12 2020

Equals -zeta(3/2)/(10*zeta(-1/2)). - Mats Granvik, May 28 2022

A122952 Decimal expansion of 3*Pi.

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Sep 30 2006

Area of the unit cycloid with cusp at the origin, whose parametric formula is x = t - sin(t) and y = 1 - cos(t).
The arc length Integral_{theta=0..2*Pi} sqrt(2(1-cos(theta))) (d theta) = 8.
3*Pi is also the surface area of a sphere whose diameter equals the square root of 3. More generally x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 18 2013
3*Pi is also the area of the nephroid (an epicycloid with two cusps) whose Cartesian parametrization is: x = (1/2) * (3*cos(t) - cos(3t)) and y = (1/2) * (3*sin(t) - sin(3t)). The length of this nephroid is 12. See the curve at the Mathcurve link. - Bernard Schott, Feb 01 2020

			9.424777960769379715387930149838508652591508198125317462924833776...

Anton, Bivens & Davis, Calculus, Early Transcendentals, 7th Edition, John Wiley & Sons, Inc., NY 2002, p. 490.
William H. Beyer, Editor, CRC St'd Math. Tables, 27th Edition, CRC Press, Inc., Boca Raton, FL, 1984, p. 214.

Robert Ferréol, Nephroid, Mathcurve.
Eric Weisstein's World of Mathematics, Cycloid.
Index entries for transcendental numbers

Cf. A000796, A019692, A019694, A019669.

Cf. A093828, A180434, A197723.

Mathematica
```
RealDigits[3Pi, 10, 111][[1]]
```

3*Pi \\ Charles R Greathouse IV, Sep 28 2022

A222362 Decimal expansion of the ratio of the area of the latus rectum segment of any equilateral hyperbola to the square of its semi-axis: sqrt(2) - log(1 + sqrt(2)).

Original entry on oeis.org

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

Offset: 0

Sylvester Reese and Jonathan Sondow, Mar 01 2013

Just as circles are ellipses whose semi-axes are equal (and are called the radius of the circle), equilateral (or rectangular) hyperbolas are hyperbolas whose semi-axes are equal.
Just as the ratio of the area of a circle to the square of its radius is always Pi, the ratio of the area of the latus rectum segment of any equilateral hyperbola to the square of its semi-axis is the universal equilateral hyperbolic constant sqrt(2) - log(1 + sqrt(2)).
Note the remarkable similarity to sqrt(2) + log(1 + sqrt(2)), the universal parabolic constant A103710, which is a ratio of arc lengths rather than of areas.  Lockhart (2012) says "the arc length integral for the parabola ... is intimately connected to the hyperbolic area integral ... I think it is surprising and wonderful that the length of one conic section is related to the area of another".
This constant is also the abscissa of the vertical asymptote of the involute of the logarithmic curve (starting point (1,0)). - Jean-François Alcover, Nov 25 2016

			0.532839975353552023569079399229905769541511547115312662423384129337355...

H. Dörrie, 100 Great Problems of Elementary Mathematics, Dover, 1965, Problems 57 and 58.
P. Lockhart, Measurement, Harvard University Press, 2012, p. 369.

G. C. Greubel, Table of n, a(n) for n = 0..10000
J.-F. Alcover, Asymptote of the logarithmic curve involute.
I.N. Bronshtein, Handbook of Mathematics, 5th ed., Springer, 2007, p. 202, eq. (3.338a).
Steven R. Finch, Mathematical Constants, Errata and Addenda, 2012, section 8.1.
J. Pahikkala, Arc Length Of Parabola, PlanetMath.
Sylvester Reese and Jonathan Sondow, Universal Parabolic Constant, MathWorld.
Eric Weisstein's World of Mathematics, Rectangular hyperbola.
Wikipedia, Equilateral hyperbola.
Wikipedia, Universal parabolic constant.
Index entries for transcendental numbers.

Cf. A002193, A091648, A103710, A103711, A180434, A278386.

Sqrt(2) - Log(Sqrt(2)+1); // G. C. Greubel, Feb 02 2018

Digits:=100: evalf(sqrt(2)-arcsinh(1)); # Wesley Ivan Hurt, Nov 27 2016

RealDigits[Sqrt[2] - Log[1 + Sqrt[2]], 10, 111][[1]]

sqrt(2)-log(sqrt(2)+1) \\ Charles R Greathouse IV, Apr 18 2013

sqrt(2)-asinh(1) \\ Charles R Greathouse IV, Dec 04 2020

Sqrt(2) - arcsinh(1), also equals Integral_{1..oo} 1/(x^2*(1+x)^(1/2)) dx. - Jean-François Alcover, Apr 16 2015

Equals Integral_{x = 0..1} x^2/sqrt(1 + x^2) dx. - Peter Bala, Feb 28 2019

A336266 Decimal expansion of (3/16)*Pi.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Jul 15 2020

Area of one egg of the "double egg" whose polar equation is r(t) = a * cos(t)^2 and a Cartesian equation is (x^2+y^2)^3 = a^2*x^4 is equal to (3/16)*Pi * a^2. See the curve at the Mathcurve link.

			0.58904862254808623221174563436490679078696926...

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 586.

Peter Bala, A note on A336266
Robert Ferréol, Double egg, Mathcurve.
Index to sequences related to curves.
Index entries for transcendental numbers

Cf. A259830 (length of an egg).

Cf. A019692 (2*Pi for deltoid), A122952 (3*Pi for cycloid and nephroid), A180434 (2-Pi/2 for Newton strophoid), A197723 (3*Pi/2 for cardioid), A336308.

Maple
```
evalf(3*Pi/16,140);
```

RealDigits[3*Pi/16, 10, 100][[1]] (* Amiram Eldar, Jul 15 2020 *)

PARI
```
3*Pi/16 \\ Michel Marcus, Jul 15 2020
```

Equals Integral_{t=0..Pi} (1/2) * cos(t)^4 * dt.
Equals Integral_{x=0..oo} 1/(x^2 + 1)^3 dx. - Amiram Eldar, Aug 13 2020
From Peter Bala, Mar 21 2024: (Start)
Equals 1/2 + Sum_{n >= 0} (-1)^n/(u(n)*u(-n)), where the polynomial u(n) = (2*n - 1)*(4*n^2 - 4*n + 3)/3 = A057813(n-1) has its zeros on the vertical line Re(z) = 1/2 in the complex plane. Cf. A336308.
Equals 1/2 + 1/(11 + 3/(12 + 15/(12 + 35/(12 + ... + (4*n^2 - 1)/(12 + ... ))))). See Lorentzen and Waadeland, p. 586, equation 4.7.10 with n = 2. (End)

A180433 Binary string formed from the binary expansion of Pi by exchanging 0's and 1's.

Original entry on oeis.org

0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0

Offset: 1

Jonathan Vos Post, Sep 05 2010

Binary complement of A004601.

May be converted to a decimal number in several ways, depending on where the binary point is inserted: .0011011011110... = 1-Pi/4, 0.011011011110... = 2-Pi/2 (see A180434), 00.11011011110... = 4-Pi.

Tarun Agarwal, Table of n, a(n) for n = 1..999

Cf. A004601, A180434.

RealDigits[Pi,2,120][[1]]/.{1->0,0->1} (* Harvey P. Dale, Apr 02 2015 *)

Verified and edited by N. J. A. Sloane, Sep 28 2010

A336308 Decimal expansion of (5/32)*Pi.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Jul 17 2020

(5*Pi/32)*a^2 is the area of a simple folium also called ovoid, or Kepler egg whose polar equation is r = a*cos^3(t) and Cartesian equation is (x^2+y^2)^2 = a * x^3. See the curve at the Mathcurve link.

			0.4908738521234051935097880286374223256558077186523602...

Robert Ferréol, Simple folium, Mathcurve.
Index to sequences related to curves.
Index entries for transcendental numbers

Cf. A019692 (2*Pi for deltoid), A122952 (3*Pi for cycloid and nephroid), A180434 (2-Pi/2 for Newton strophoid), A197723 (3*Pi/2 for cardioid), A336266 (3*Pi/16 for double egg).

Maple
```
evalf(5*Pi/32, 140);
```

RealDigits[5*Pi/32, 10, 100][[1]] (* Amiram Eldar, Jul 17 2020 *)

PARI
```
5*Pi/32 \\ Michel Marcus, Jul 17 2020
```

Equals Integral_{t=0..Pi} cos^6(t)/2 dt (area of simple folium).
From Amiram Eldar, Aug 13 2020: (Start)
Equals Integral_{x=0..oo} 1/(x^2 + 1)^4 dx.
Equals Integral_{x=-1..1} x^3 * arcsin(x) dx. (End)
Equals 5/9 - 10*Sum_{n >= 1} (-1)^(n+1)/(u(n)*u(-n)), where the polynomial u(n) = (2*n - 1)^2*(4*n^2 - 4*n + 9)/3 satisfies the difference equation 16*u(n) = (2*n - 1)*(u(n+1) - u(n-1)) and has its zeros on the vertical line Re(z) = 1/2 in the complex plane. Cf. A336266. - Peter Bala, Mar 25 2024

A237841 Decimal expansion of Ramanujan's AGM Continued Fraction R(2) = R_1(2,2).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Feb 14 2014

Other closed form evaluations of R(p/q):
R(1/4) = Pi/2 - 4/3,
R(1/3) = 1 - log(2),
R(1/2) = 2 - Pi/2,
R(2/3) = 4 - Pi/sqrt(2),
R(1) = log(2),
R(3/2) = Pi + sqrt(3)*log(2 - sqrt(3)),
R(3) = Pi/sqrt(3) - log(2).

			0.97499098879872209671990033452921084400592...

D. H. Bailey, J. M. Borwein, V. Kapoor and E. Weisstein, Ten Problems in Experimental Mathematics, page 13.
Jonathan M. Borwein, Ramanujan's Arithmetic-Geometric Mean Continued Fractions and Dynamics

Cf. A002162: R(1) = log(2); A180434: R(1/2) = 2-Pi/2.

RealDigits[Sqrt[2]*(Pi/2 - Log[1 + Sqrt[2]]), 10, 100] // First

(psi(7/8)-psi(3/8))/2 \\ Charles R Greathouse IV, Mar 03 2016

Equivalent formulas:
sqrt(2)*(Pi/2 - log(1 + sqrt(2))),
(Pi - 2*arccoth(sqrt(2)))/sqrt(2),
Integral_{x >= 0} sech(Pi*x/4)/(1 + x^2) dx,
2*Integral_{x = 0..1} sqrt(x)/(1 + x^2) dx,
Integral_{x >= 0} exp(-x/2)*sech(x) dx,
4*Sum_{k >= 1} (-1)^(k+1)/(4*k - 1),
1/2*(-psi(3/8) + psi(7/8)), where psi is the digamma function,
4/3 * 2F1(3/4, 1, 7/4, -1), where 2F1 is the hypergeometric function,
(H(-1/8) - H(-5/8))/2, where H(n) is the n-th harmonic number.
General formula:
The Borwein's closed form formula for R(n) with n integer simplifies to:
R(n) = Pi/2*sec(Pi/(2n)) - 2*sum( cos((k*(n+1)*Pi)/(2*n))*log(2*sin((k*Pi)/(4*n))), {k, 1, 2n-1, 2} ).
Equals 4*A181049. - Peter Bala, Apr 02 2024

A351055 Decimal expansion of sqrt(2 - Pi/2), the coefficient for standard distribution of the Rayleigh random variable.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, Feb 07 2022

The standard distribution of a Rayleigh random variable with shape parameter s is s*sqrt(2 - Pi/2).

			0.65513637756203355309393588562466171054101828519553...

Index entries for transcendental numbers

Cf. A000796, A019669, A180434.

RealDigits[Sqrt[2 - Pi/2], 10, 100][[1]] (* Amiram Eldar, Feb 07 2022 *)

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

Equals sqrt(A180434).

A359104 Decimal expansion of the area enclosed by Sylvester's Bicorn curve.

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Dec 18 2022

The Cartesian equation of Sylvester's Bicorn curve is y^2*(m^2-x^2) = (x^2+2*m*y-m^2)^2, here with parameter m=1. The area is proportional to the square m^2 of parameter m.

Corresponding arc length is given by A228764.

			0.746455945439346463341461672758965758770535375107896820343...

M. Protat, Des Olympiades à l'Agrégation, Encadrement du bicorne, Problème 66, pp. 142-145, Ellipses, Paris 1997.

Robert Ferréol, Bicorn, Mathcurve.
Eric Weisstein's World of Mathematics, Bicorn.
Wikipedia, Bicorn.
Index to sequences related to curves.

Cf. A228764 (length).

Other area of curves: A019692 (deltoid), A197723 (cardioid), A122952 (nephroid), A180434 (Newton strophoid).

Maple
```
evalf((16*sqrt(3) - 27)*Pi/3, 100);
```

RealDigits[(16*Sqrt[3] - 27)*Pi/3, 10, 120][[1]] (* Amiram Eldar, Dec 18 2022 *)

Equals (16*sqrt(3) - 27)*Pi/3.

cp's OEIS Frontend

A019692 Decimal expansion of 2*Pi.

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A019694 Decimal expansion of 2*Pi/5.

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A122952 Decimal expansion of 3*Pi.

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A222362 Decimal expansion of the ratio of the area of the latus rectum segment of any equilateral hyperbola to the square of its semi-axis: sqrt(2) - log(1 + sqrt(2)).

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A336266 Decimal expansion of (3/16)*Pi.

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A180433 Binary string formed from the binary expansion of Pi by exchanging 0's and 1's.

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