A249491 -id:A249491 - cp's OEIS frontend

A093728 Decimal expansion of 2E(2i sqrt(2)), where E(k) is the complete elliptic integral of the 2nd kind.

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

Offset: 1

Eric W. Weisstein, Apr 13 2004

Arc length of the trifolium r = a*cos(3*theta).

			6.68244661027762911506475116253009811970049133619458855164648...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Trifolium

Cf. A249491.

Re(evalf(2*EllipticE(2*I*sqrt(2)), 120)); # Vaclav Kotesovec, Apr 22 2015

RealDigits[ N[ 2*EllipticE[-8], 102]][[1]] (* Jean-François Alcover, Oct 29 2012 *)

2*intnum(t=0,Pi/2,sqrt(1+8*sin(t)^2)) \\ Charles R Greathouse IV, Aug 15 2015

Equals 2*A249491. - 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.

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.

A334848 Decimal expansion of circumference of x^2 + 9 y^2 = 9.

Original entry on oeis.org

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

Offset: 2

Clark Kimberling, Jun 15 2020

			arclength = 13.3648932205552582301295023250601962394009826...

Cf. A093728, A249491, A333202, A334849.

s = Integrate[Sqrt[D[ 3 Cos[t], t]^2 + D[Sin[t], t]^2], {t, 0, 2 Pi}]
r = N[s, 200]
RealDigits[r][[1]]

arclength = 4*E(-8), where E = complete elliptic integral.

Equals 2*A093728 = 4*A249491.

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

A093728 Decimal expansion of 2E(2i sqrt(2)), where E(k) is the complete elliptic integral of the 2nd kind.

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

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

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A334848 Decimal expansion of circumference of x^2 + 9 y^2 = 9.

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

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Mathematica

Formula

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

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula