OFFSET
0,1
COMMENTS
Example of extension to Buffon's Needle Problem: The probability that the boundary of a square will intersect one of the parallel lines if the square's diagonal length l (almost) equals the distance d between each pair of lines. This follows directly from the Weisstein/MathWorld Buffon's Needle Problem link's statement P=p/(Pi*d), where P is the probability of intersection with any convex polygon's boundary if the generalized diameter of that polygon is less than d and p is the perimeter of the polygon. (Take d=l, then p=2*sqrt(2)*d.).
The area of a regular octagon circumscribed in a unit-area circle. - Amiram Eldar, Nov 05 2020
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Buffon's needle problem.
Eric Weisstein's World of Mathematics, Generalized Diameter.
FORMULA
Equals Product_{n>=1} (1-1/(4*n)^2). - Bruno Berselli, Apr 02 2013
Equals sinc(Pi/4). - Peter Luschny, Oct 04 2019
Equals Product_{k>=3} cos(Pi/2^k). - Amiram Eldar, Aug 24 2020
EXAMPLE
0.9003163161571060695551991910067405826645741499552206255714374712314587307...
MATHEMATICA
RealDigits[2 Sqrt[2]/Pi, 10, 110][[1]] (* Bruno Berselli, Apr 02 2013 *)
(* From the second comment: *) RealDigits[N[Product[1 - 1/(4 n)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)
PROG
(PARI) 2*sqrt(2)/Pi
(Magma) R:= RealField(100); 2*Sqrt(2)/Pi(R); // G. C. Greubel, Aug 17 2018
CROSSREFS
KEYWORD
cons,nonn
AUTHOR
Rick L. Shepherd, Jan 11 2006
STATUS
approved