A348681 - OEIS

%I #6 Oct 29 2021 09:05:50

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

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

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

%N Decimal expansion of the average length of a chord in a unit square defined by the intersection of the perimeter with a straight line passing through 2 points uniformly and independently chosen at random in the interior of the square.

%D A. M. Mathai, An introduction to geometrical probability: distributional aspects with applications, Amsterdam: Gordon and Breach, 1999, p. 221, ex. 2.3.7.

%H Rodney Coleman, <a href="https://www.jstor.org/stable/3212012">Random paths through convex bodies</a>, Journal of Applied Probability, Vol. 6, No. 2 (1969), pp. 430-441; <a href="https://doi.org/10.2307/3212012">alternative link</a>; <a href="https://www.researchgate.net/publication/268246373_Random_Paths_Through_Convex_Bodies">author's link</a>.

%F Equals (2/3) * (log(1 + sqrt(2)) + (2 + sqrt(2))/5).

%e 1.04281086632944135666196471321448794982806313555801...

%t RealDigits[(2/3) * (Log[1 + Sqrt[2]] + (2 + Sqrt[2])/5), 10, 100][[1]]

%o (PARI) (2/3) * (log(1 + sqrt(2)) + (2 + sqrt(2))/5) \\ _Michel Marcus_, Oct 29 2021

%Y Cf. A091505, A247674, A348680, A348682, A348683.

%K nonn,cons

%O 1,3

%A _Amiram Eldar_, Oct 29 2021