A232715 -id:A232715 - cp's OEIS frontend

A177870 Decimal expansion of 3*Pi/4.

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

Offset: 1

R. J. Mathar, Dec 13 2010

As radians, this is equal to 135 degrees (on an analog clock, the span of 22 minutes and 30 seconds). - Alonso del Arte, Feb 03 2013
Ratio of the area of an arbelos to the area of its associated parbelos. - Jonathan Sondow, Nov 28 2013
(3*Pi/4)*a^2 is the area between a cissoid of Diocles and its asymptote when polar equation of cissoid is r = a*sin^2(t)/cos(t) and Cartesian equation is x * (x^2+y^2) = a * y^2 or y = +-x * sqrt(x/(a-x)). See the curve at the Mathcurve link and formula. - Bernard Schott, Jul 14 2020
The smallest nonnegative solution to sin(x) = -cos(x). - Wolfe Padawer, Apr 12 2023

			2.35619449019234492884698253745962716314787704953132936573120...

Jonathan Borwein & Peter Borwein, A Dictionary of Real Numbers. Pacific Grove, California: Wadsworth & Brooks/Cole Advanced Books & Software (1990) p. 168

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Robert Ferréol, Cissoid of Diocles, Mathcurve.
Jonathan Sondow, The parbelos, a parabolic analog of the arbelos, arXiv:1210.2279 [math.HO], 2012-2013: Amer. Math. Monthly 120 (2013) 929-935.
E. Tsukerman, Solution of Sondow's problem: a synthetic proof of the tangency property of the parbelos, arXiv:1210.5580 [math.MG], 2012-2013.
Index to sequences related to curves
Index entries for transcendental numbers

Reciprocal of A232715.

Cf. A000796, A003881, A019675, A122952.

Maple
```
evalf(3*Pi/4) ;
```
Mathematica
```
RealDigits[N[3(Pi/4), 110]][[1]]
```

3*Pi/4 \\ Charles R Greathouse IV, Sep 30 2022

Equals (3/4)*A000796 = 3*A003881 = 6*A019675 = A122952/4.
Equals 1 + (3/5) + (3*4)/(5*7) + (3*4*5)/(5*7*9) + ... = hypergeom([3,1],[5/2],1/2). - Peter Bala, Oct 30 2019
Equals 2 * Integral_{x=0..1} x * sqrt(x/(1-x)) dx (cissoid). - Bernard Schott, Jul 14 2020
Equals Sum_{k>=1} arctan(2/k^2). - Amiram Eldar, Aug 10 2020
Equals Integral_{x=-oo..oo} (x^2 + 1)/(x^4 + 4) dx. - Kritsada Moomuang, Jun 04 2025

A232717 Decimal expansion of the ratio of the length of the boundary of any arbelos to the length of the boundary of its associated parbelos: Pi / (sqrt(2) + log(1 + sqrt(2))).

Original entry on oeis.org

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

Offset: 1

Jonathan Sondow, Nov 28 2013

Same as decimal expansion of Pi/P, where P is the Universal parabolic constant (A103710). - Jonathan Sondow, Jan 19 2015

			1.36853556373191478886062626593258810842142480010621734905399...

G. C. Greubel, Table of n, a(n) for n = 1..10000
M. Hajja, Review Zbl 1291.51018, zbMATH 2015.
M. Hajja, Review Zbl 1291.51016, zbMATH 2015.
J. Sondow, The parbelos, a parabolic analog of the arbelos, arXiv 2012, Amer. Math. Monthly, 120 (2013), 929-935.
E. Tsukerman, Solution of Sondow's problem: a synthetic proof of the tangency property of the parbelos, arXiv 2012, Amer. Math. Monthly, 121 (2014), 438-443.

Reciprocal of A232716. Ratio of areas is A232715.

Cf. A000796, A103710.

R:= RealField(); Pi(R)/(Sqrt(2) + Log(1 + Sqrt(2))) // G. C. Greubel, Jul 27 2018

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

Pi/(sqrt(2) + log(1 + sqrt(2))) \\ G. C. Greubel, Jul 27 2018

Equals A000796 / A103710.

A301755 Decimal expansion of 3/8.

Original entry on oeis.org

3, 7, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Natan Arie Consigli, Mar 26 2018

This is the value of Zc which is the critical compression factor for Van der Waals gases.
The compression factor of a gas is defined as Z = p Vm/RT, where p is the pressure, Vm is the molar volume, T the temperature and R the gas constant.
The critical forms for Van der Waals gasses of pressure, temperature and volume are respectively: pc = a/(27b^2), Tc = 8a/(27R*b), Vc = 3b.
By plugging in the critical forms of the variables you will get Zc = pc Vc/RTc = 3/8.
Probability of intersection of 2 random chords in a circle, when their midpoints are selected randomly in the circle. See Solomon. - Michel Marcus, Mar 29 2018
The distance of the centroid of a unit-radius solid hemisphere from its base (the 3D analog of A232715). - Amiram Eldar, Jul 19 2022
The binomial transform is 3, 10, 22, 39,... A140066 without the leading 1. - R. J. Mathar, Feb 15 2023

			0.3750000...

Peter Atkins, Julio de Paula, Physical Chemistry 8th Ed., W. H. Freeman and Company, 2006, pp. 14-21.

Herbert Solomon, Geometric Probability, SIAM, 1978, p. 152.
Wikipedia, Compressibility factor.
Wikipedia, Van der Waals equation.

Cf. A232715.

First@ RealDigits[N[3/8, 105]] (* Michael De Vlieger, May 15 2018 *)

my(x=3/8); for(k=1, 100, my(d=floor(x)); x=(x-d)*10; k>1 && print1(d, ", ")) \\ Felix Fröhlich, Mar 28 2018

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A177870 Decimal expansion of 3*Pi/4.

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A232717 Decimal expansion of the ratio of the length of the boundary of any arbelos to the length of the boundary of its associated parbelos: Pi / (sqrt(2) + log(1 + sqrt(2))).

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A301755 Decimal expansion of 3/8.

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