A102559 -id:A102559 - cp's OEIS frontend

A102557 Denominator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle.

4, 32, 512, 4096, 131072, 1048576, 16777216, 134217728, 8589934592, 68719476736, 1099511627776, 8796093022208, 281474976710656, 2251799813685248, 36028797018963968, 288230376151711744, 36893488147419103232, 295147905179352825856, 4722366482869645213696, 37778931862957161709568

Offset: 1

Eric W. Weisstein, Jan 14 2005

Presumably this is the same as A093581? - Andrew S. Plewe, Apr 18 2007

a(n) equals A093581(n) for n <= 55000. - G. C. Greubel, Oct 20 2024

			3/4, 15/32, 159/512, 867/4096, 19239/131072, 107985/1048576, ...

Robert Israel, Table of n, a(n) for n = 1..830
Eric Weisstein's World of Mathematics, Gaussian Triangle Picking

Cf. A093581, A102556, A102558, A102559, A378071.

A102557:= func< n | Power(2, 4*n-2-(&+Intseq(2*(n-1), 2))) >;
[A102557(n): n in [1..30]]; // G. C. Greubel, Oct 20 2024

p:= gfun:-rectoproc({(-6*n-3)*v(n)+(14*n+11)*v(n+1)+(-8*n-8)*v(n+2), v(0) = 0, v(1) = 3/4, v(2) = 15/32},v(n),remember):
seq(denom(p(n)),n=1..50); # Robert Israel, Sep 29 2016
# Alternative, assuming offset 0:
a := n -> numer((4^(2*n+1)*n!^2)/((2*n+3)*(2*n)!)): # Peter Luschny, Dec 05 2024

a[n_]:= (3^n/4^(2n-1)) Binomial[2n-1, n] Hypergeometric2F1[1, 1-n, 1+n, -1/3] // Denominator; Array[a, 20] (* Jean-François Alcover, Mar 22 2019 *)

a(n) = denominator(sum(k=n, 2*n-1, binomial(2*n-1,k)*3^(2*n-k)/4^(2*n-1))); \\ Michel Marcus, Mar 23 2019

def A102557(n): return pow(2, 4*n-2 - sum((2*n-2).digits(2)))
[A102557(n) for n in range(1,31)] # G. C. Greubel, Oct 20 2024

From Robert Israel, Sep 29 2016: (Start)
a(n) is the denominator of p(n) = Sum_{k=n..2n-1} binomial(2n-1,k) 3^(2n-k)/4^(2n-1).
8*(n+1)*p(n+2) = (14*n+11)*p(n+1) - 3*(2*n+1)*p(n), for n >= 1, with p(0) = 0, p(1) = 3/4, and p(2) = 15/32.
G.f. of p(n): 3*x*(1 - 1/sqrt(4-3*x))/(2*(1-x)). (End)
Assuming offset 0: a(n) = numerator((4^(2*n+1)*n!^2)/((2*n+3)*(2*n)!)). - Peter Luschny, Dec 05 2024

A102556 Numerator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle.

Original entry on oeis.org

3, 15, 159, 867, 19239, 107985, 1222563, 6965835, 319153335, 1835486085, 21185534577, 122622340677, 2846090375067, 16550504577861, 192854402926251, 1125503935556763, 105252693980913879, 615999836125850637, 7219077361263238917, 42347454581722163361, 994637701798929524937

Offset: 1

Eric W. Weisstein, Jan 14 2005

			p(n) = {3/4, 15/32, 159/512, 867/4096, 19239/131072, 107985/1048576, ... }_{n >= 1}.

Robert Israel, Table of n, a(n) for n = 1..928
Eric Weisstein's World of Mathematics, Gaussian Triangle Picking

Cf. A102557 (denominators), A102558, A102559.

R:=PowerSeriesRing(Rationals(), 50);
A102556:= func< n | Numerator( Coefficient(R!( 3*x*(1-1/Sqrt(4-3*x))/(2-2*x) ), n) ) >;
[A102556(n): n in [1..30]]; // G. C. Greubel, Jan 31 2025

p:= gfun:-rectoproc({(-6*n-3)*v(n)+(14*n+11)*v(n+1)+(-8*n-8)*v(n+2), v(0) = 0, v(1) = 3/4, v(2) = 15/32},v(n),remember):
seq(numer(p(n)),n=1..50); # Robert Israel, Sep 29 2016

a[n_] := (3^n/4^(2n-1)) Binomial[2n-1, n] Hypergeometric2F1[1, 1-n, 1+n, -1/3] // Numerator; Array[a, 20] (* Jean-François Alcover, Mar 22 2019 *)

a(n) = numerator(sum(k=n, 2*n-1, binomial(2*n-1,k)*3^(2*n-k)/4^(2*n-1))); \\ Michel Marcus, Mar 23 2019

def A102556(n): return ( 3*(1-1/sqrt(4-3*x))/(2*(1-x)) ).series(x,n+1).list()[n].numerator()
print([A102556(n) for n in range(31)]) # G. C. Greubel, Jan 31 2025

From Robert Israel, Sep 29 2016: (Start)
a(n) is the numerator of p(n) = Sum_{k=n..2*n-1} binomial(2*n-1,k)*3^(2*n-k)/4^(2*n-1).
8(n+1)*p(n+2) = (14*n+11)*p(n+1) - 3*(2*n+1)*p(n), for n >= 1.
G.f. of p(n):  3*x*(1 - 1/sqrt(4-3*x))/(2-2*x). (End)

A102558 Numerator of the probability that (2n+1)-dimensional Gaussian random triangle has an obtuse angle.

Original entry on oeis.org

3, 9, 27, 837, 891, 729, 12393, 277749, 4782969, 91703097, 92293587, 82019061, 2674388259, 10722885057, 155747819547, 19336668383673, 667382013477, 1019303306559, 716912704223253, 717162977859147, 29411190301301847

Offset: 1

Eric W. Weisstein, Jan 14 2005

			1 - (3*sqrt(3))/(4*Pi), 1 - (9*sqrt(3))/(8*Pi), 1 - (27*sqrt(3))/(20*Pi), ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Gaussian Triangle Picking

Cf. A102556, A102557, A102559 (denominator).

Table[Numerator[Simplify[Pi/Sqrt[3] - 3^(n+1)*Hypergeometric2F1[1/2, 1/2 + n, 3/2+n, 3/4]/(2*(2*n+1)*Binomial[2*n,n])]], {n,40}] (* G. C. Greubel, Feb 01 2025 *)

From G. C. Greubel, Feb 01 2025: (Start)
a(n) = numerator( p(n) ), where p(n) = Pi/sqrt(3) - (3^(n+1)/(2*binomial(2*n, n))) * Sum_{k>=0} binomial(2*k, k)*(3/16)^k/(2*k + 2*n + 1).
a(n) = numerator( p(n) ), where p(n) = Pi/sqrt(3) - (3^(n+1)/(2*(2*n+1)*binomial(2*n,n))) * Hypergeometric2F1([1/2, 1/2 + n], [3/2+n], 3/4). (End)

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

A102557 Denominator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle.

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A102556 Numerator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle.

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A102558 Numerator of the probability that (2n+1)-dimensional Gaussian random triangle has an obtuse angle.

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

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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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Maple

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