A102520 -id:A102520 - cp's OEIS frontend

A102519 Decimal expansion of 1-(3sqrt(3))/(4Pi).

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

Offset: 0

Mark Hudson (mrmarkhudson(AT)hotmail.com), Jan 13 2005

This is the probability that a Gaussian triangle in 3 dimensions is obtuse.

Also the probability that the distance between 2 randomly selected points within a circle will be smaller than the radius. - Amiram Eldar, Mar 03 2019

			0.58650332843365596286650512626527291895196013972501951040047548478172727...

G. C. Greubel, Table of n, a(n) for n = 0..5000
M. W. Crofton, Problem 1829, solved by J. J. Sylvester and by the proposer, Mathematical Questions with Their Solutions: From the "Educational Times", Vol. 4 (1866), pp. 77-78.
S. R. Finch, Random Triangles, Jan 21 2010. [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Gaussian Triangle Picking
Index entries for transcendental numbers

Cf. A102520, A240935.

RealDigits[1 - (3*Sqrt[3])/(4*Pi), 10, 50][[1]] (* G. C. Greubel, Jun 02 2017 *)

1 - (3*sqrt(3))/(4*Pi) \\ G. C. Greubel, Jun 02 2017

A249491 Decimal expansion of the expected product of two sides of a random Gaussian triangle (in two dimensions).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 30 2014

Coordinates are independent normally distributed random variables with mean 0 and variance 1.

			3.341223305138814557532375581265049059850245668...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Steven R. Finch, Random Triangles, Jan 21 2010. [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Gaussian Triangle Picking

Cf. A093728, A102519, A102520, A102556, A102557, A102558, A102559.

Re(evalf(4*EllipticE(1/2)-sqrt(3)*EllipticK(I/sqrt(3)), 120)); # Vaclav Kotesovec, Apr 22 2015

ek[x_] := EllipticK[x^2/(-1 + x^2)]/Sqrt[1 - x^2]; ee[x_] := EllipticE[x^2]; p = 4*ee[1/2] - (3/2)*ek[1/2]; (* or *) p = 4*EllipticE[1/4] - Sqrt[3]*EllipticK[-1/3]; RealDigits[p, 10, 104] // First
RealDigits[ N[ EllipticE[-8], 102]][[1]] (* Altug Alkan, Oct 02 2018 *)
RealDigits[3 EllipticE[8/9], 10, 102][[1]] (* Jan Mangaldan, Nov 24 2020 *)

magm(a,b)=my(eps=10^-(default(realprecision)-5), c); while(abs(a-b)>eps, my(z=sqrt((a-c)*(b-c))); [a,b,c] = [(a+b)/2,c+z,c-z]); (a+b)/2
E(x)=Pi/2/agm(1,sqrt(1-x))*magm(1,1-x)
K(x)=Pi/2/agm(1,sqrt(1-x))
4*E(1/4) - sqrt(3)*K(-1/3) \\ Charles R Greathouse IV, Aug 02 2018

p = 4*E(1/4) - sqrt(3)*K(-1/3), where E is the complete elliptic integral and K the complete elliptic integral of the first kind.

Equals A093728/2. - Altug Alkan, Oct 02 2018

A249492 Decimal expansion of rho(a,b), the cross-correlation coefficient of two sides of a random Gaussian triangle (in two dimensions).

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 30 2014

Coordinates are independent normally distributed random variables with mean 0 and variance 1.

			0.23255934654317823447309035975033389931043501543502...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Steven R. Finch, Random Triangles, January 21, 2010. [Cached copy, with permission of the author]
Eric Weisstein MathWorld, Gaussian Triangle Picking

Cf. A102519, A102520, A102556, A102557, A102558, A102559, A249491.

Re(evalf((4*EllipticE(1/2) - sqrt(3)*EllipticK(I/sqrt(3)) - Pi)/(4 - Pi), 120)); # Vaclav Kotesovec, Apr 22 2015

p = 4*EllipticE[1/4] - Sqrt[3]*EllipticK[-1/3]; rho = (p - Pi)/(4 - Pi); RealDigits[rho, 10, 103] // First
RealDigits[(3 EllipticE[8/9] - Pi)/(4 - Pi), 10, 103][[1]] (* Jan Mangaldan, Nov 26 2020 *)

rho = (p - Pi)/(4 - Pi), where p is A249491, the expected value of the product of two sides.

A249539 Decimal expansion of 12/sqrt(Pi), the average perimeter of a random Gaussian triangle in three dimensions.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 31 2014

			6.770275002573075443376953418729271030128607551947986282129...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Steven R. Finch, Random Triangles, Jan 21 2010. [Cached copy, with permission of the author]
Eric Weisstein's MathWorld, Gaussian Triangle Picking
Index entries for transcendental numbers

Cf. A102519, A102520, A102558, A102559, A249521 (average side length in three dimensions), A249538 (average perimeter in two dimensions).

RealDigits[12/Sqrt[Pi], 10, 105] // First

12/sqrt(Pi) \\ Charles R Greathouse IV, Apr 20 2016

A249542 Decimal expansion of the average product of a side and an adjacent angle of a random Gaussian triangle in two dimensions.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 31 2014

Coordinates are independent normally distributed random variables with mean 0 and variance 1.
As of 2010, an exact expression of this constant was not known, according to Steven Finch.
This average product is noticeably smaller than the product of the averages sqrt(Pi)*Pi/3 = 1.8561..., the side length being negatively correlated with the adjacent angle value.

			1.6377293248568680327801569567984764558203819870905934...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Steven R. Finch, Random Triangles, January 21, 2010, p. 14. [Cached copy, with permission of the author]
Eric Weisstein's MathWorld, Gaussian Triangle Picking

Cf. A102519, A102520, A102556, A102557, A102558, A102559, A249491, A249492, A249538.

ex = (-Sqrt[3]*Log[3] + Pi^2 - 8*PolyLog[2, 2-Sqrt[3]] + 2*PolyLog[2, 7-4*Sqrt[3]])/(2*Sqrt[Pi]); RealDigits[ex, 10, 105] // First

from mpmath import *
mp.dps=106
C = (-sqrt(3)*log(3) + pi**2 - 8*polylog(2, 2-sqrt(3)) + 2*polylog(2, 7 - 4*sqrt(3)))/(2*sqrt(pi))
print([int(n) for n in list(str(C).replace('.', '')[:-1])]) # Indranil Ghosh, Jul 04 2017

Equals (1/(3*Pi))*Integral_{x=0..oo} Integral_{y=0..oo} Integral_{t=0..Pi} x^2*y*t*exp(-(1/3)*(x^2 - x*y*cos(t) + y^2)) dt dy dx.

Equals (-sqrt(3)*log(3) + Pi^2 - 8*Li_2(2-sqrt(3)) + 2*Li_2(7-4*sqrt(3)))/(2*sqrt(Pi)), where Li_2 is the dilogarithm function.

A249522 Decimal expansion of the expected product of two sides of a random Gaussian triangle in three dimensions.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 31 2014

			5.3079733725307522970679589898778166483843188821998439167961...

Steven R. Finch, Random Triangles, January 21, 2010. [Cached copy, with permission of the author]
Eric Weisstein MathWorld, Gaussian Triangle Picking

Cf. A102519, A102520, A102556, A102557, A102558, A102559, A249491, A249521.

p = 2 + 6*Sqrt[3]/Pi; RealDigits[p, 10, 104] // First

p = 2 + 6*sqrt(3)/Pi.

A249523 Decimal expansion of rho(a,b), the cross-correlation coefficient of two sides of a random Gaussian triangle in three dimensions.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 31 2014

Coordinates are independent normally distributed random variables with mean 0 and variance 1.

			0.237051025209600286585569504676408571940176916236978...

Steven R. Finch, Random Triangles, January 21, 2010. [Cached copy, with permission of the author]
Eric Weisstein's MathWorld, Gaussian Triangle Picking

Cf. A102519, A102520, A102556, A102557, A102558, A102559, A249491, A249492.

rho = (-8 + 3*Sqrt[3] + Pi)/(-8 + 3*Pi); RealDigits[rho, 10, 103] // First

(sqrt(27)+Pi-8)/(3*Pi-8) \\ Charles R Greathouse IV, Apr 20 2016

rho = (-8 + 3*sqrt(3) + Pi)/(-8 + 3*Pi).

cp's OEIS Frontend

A102519 Decimal expansion of 1-(3*sqrt(3))/(4*Pi).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A249491 Decimal expansion of the expected product of two sides of a random Gaussian triangle (in two dimensions).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A249492 Decimal expansion of rho(a,b), the cross-correlation coefficient of two sides of a random Gaussian triangle (in two dimensions).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A249539 Decimal expansion of 12/sqrt(Pi), the average perimeter of a random Gaussian triangle in three dimensions.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A249542 Decimal expansion of the average product of a side and an adjacent angle of a random Gaussian triangle in two dimensions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A249522 Decimal expansion of the expected product of two sides of a random Gaussian triangle in three dimensions.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A249523 Decimal expansion of rho(a,b), the cross-correlation coefficient of two sides of a random Gaussian triangle in three dimensions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A102519 Decimal expansion of 1-(3sqrt(3))/(4Pi).