A102557 -id:A102557 - cp's OEIS frontend

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

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)

A102559 Denominator of the probability that (2n+1)-dimensional Gaussian random triangle has an obtuse angle.

Original entry on oeis.org

4, 8, 20, 560, 560, 440, 7280, 160160, 2722720, 51731680, 51731680, 45762640, 1487285800, 5949143200, 86262576400, 10696559473600, 368846878400, 562976814400, 395772700523200, 395772700523200, 16226680721451200

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, A102558 (numerator).

Table[Denominator[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,30}] (* G. C. Greubel, Feb 01 2025 *)

From G. C. Greubel, Feb 01 2025: (Start)
a(n) = denominator( 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) = denominator( 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

A093581 Numerators of odd moments in the distribution of chord lengths for picked at random on the circumference of a unit circle.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Apr 01 2004

Presumably this is the same as A102557? - Andrew S. Plewe, Apr 18 2007

A102557(n) equals a(n) for n <= 55000. - G. C. Greubel, Oct 20 2024

			1, 4/Pi, 2, 32/(3*Pi), 6, 512/(15*Pi), 20, 4096/(35*Pi), ...

G. C. Greubel, Table of n, a(n) for n = 1..828
Eric Weisstein's World of Mathematics, Circle Line Picking

Cf. A000984, A001803, A061549, A102557.

Denominators are A001803*Pi.

A093581:= func< n | Power(2, 4*n-2-(&+Intseq(2*(n-1), 2))) >;
[A093581(n): n in [1..30]]; // G. C. Greubel, Oct 20 2024

Table[Power[2, 4*n-2 - DigitCount[n-1,2,1]], {n, 30}] (* G. C. Greubel, Oct 20 2024 *)

def A093581(n): return pow(2, 4*n-2 - sum((2*n-2).digits(2)))
[A093581(n) for n in range(1,31)] # G. C. Greubel, Oct 20 2024

a(n) = 4*A061549(n-1).

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.

A249538 Decimal expansion of 3*sqrt(Pi), the average perimeter of a random Gaussian triangle in two dimensions.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Oct 31 2014

			5.317361552716548081894502450023435548392648368367161...

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. A002161 (average side length in two dimensions), A102556, A102557.

RealDigits[3*Sqrt[Pi], 10, 105] // First

3*sqrt(Pi) \\ G. C. Greubel, Jan 09 2017

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.

A378071 a(n) = denominator((4^(2n+1)n!^2)/((2n+3)(2*n)!)).

Original entry on oeis.org

3, 5, 21, 45, 385, 819, 3465, 7293, 122265, 255255, 1062347, 2204475, 18253053, 37702175, 155451825, 319929885, 10518906825, 21585857535, 88482569175, 181144476975, 1481850184815, 3027700543725, 12361581411855, 25214881603275, 411156946959525, 837470267650107

Offset: 0

Peter Luschny, Dec 05 2024

Cf. A102557 (numerator).

a := n -> (4^(2*n+1)*n!^2)/((2*n+3)*(2*n)!); seq(denom(a(n)), n = 0..25);

Table[Denominator[(4^(2*n + 1)*n!^2)/((2*n + 3)*(2*n)!)], {n, 0, 25}] (* Michael De Vlieger, Dec 05 2024 *)

a(n) = denominator((4^(2*n+1)*n!^2)/((2*n+3)*(2*n)!)); \\ Michel Marcus, Dec 05 2024

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.

cp's OEIS Frontend

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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A102559 Denominator 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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A093581 Numerators of odd moments in the distribution of chord lengths for picked at random on the circumference of a unit circle.

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Magma

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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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A249538 Decimal expansion of 3*sqrt(Pi), the average perimeter of a random Gaussian triangle in two dimensions.

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A378071 a(n) = denominator((4^(2n+1)n!^2)/((2n+3)(2*n)!)).