A102558 Numerator of the probability that (2n+1)-dimensional Gaussian random triangle has an obtuse angle.
3, 9, 27, 837, 891, 729, 12393, 277749, 4782969, 91703097, 92293587, 82019061, 2674388259, 10722885057, 155747819547, 19336668383673, 667382013477, 1019303306559, 716912704223253, 717162977859147, 29411190301301847
Offset: 1
Examples
1 - (3*sqrt(3))/(4*Pi), 1 - (9*sqrt(3))/(8*Pi), 1 - (27*sqrt(3))/(20*Pi), ...
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Eric Weisstein's World of Mathematics, Gaussian Triangle Picking
Programs
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Mathematica
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 *)
Formula
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)