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A102557
Denominator of the probability that 2n-dimensional Gaussian random triangle has an obtuse angle.
11
4, 32, 512, 4096, 131072, 1048576, 16777216, 134217728, 8589934592, 68719476736, 1099511627776, 8796093022208, 281474976710656, 2251799813685248, 36028797018963968, 288230376151711744, 36893488147419103232, 295147905179352825856, 4722366482869645213696, 37778931862957161709568
OFFSET
1,1
COMMENTS
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
LINKS
Eric Weisstein's World of Mathematics, Gaussian Triangle Picking
FORMULA
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)
EXAMPLE
3/4, 15/32, 159/512, 867/4096, 19239/131072, 107985/1048576, ...
MAPLE
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
MATHEMATICA
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 *)
PROG
(PARI) 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
(Magma)
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
(SageMath)
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
CROSSREFS
KEYWORD
nonn,frac
AUTHOR
Eric W. Weisstein, Jan 14 2005
STATUS
approved