A348670 - OEIS

%I #19 Feb 07 2024 15:33:54

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

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

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

%N Decimal expansion of 10 - Pi^2.

%C Let ABC be a unit-area triangle, and let P be a point uniformly picked at random inside it. Let D, E and F be the intersection points of the lines AP, BP and CP with the sides BC, CA and AB, respectively. Then, the expected value of the area of the triangle DEF is this constant.

%D Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013, p. 220.

%D A. M. Mathai, An introduction to geometrical probability: distributional aspects with applications, Amsterdam: Gordon and Breach, 1999, p. 275, ex. 2.5.3.

%H R. W. Gosper, <a href="https://dspace.mit.edu/handle/1721.1/6088">Acceleration of Series</a>, AIM-304 (1974), page 71.

%H Olivier Schneegans, <a href="https://doi.org/10.1080/00029890.2019.1577102">How Close to 10 is Pi^2?</a>, The American Mathematical Monthly, Vol. 126, No. 5 (2019), p. 448.

%H Daniel Sitaru, <a href="https://cms.math.ca/publications/crux/issue/?volume=49&amp;issue=7">Problem B131</a>, Crux Mathematicorum, Vol. 49, No. 7 (2023), p. 381.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals Sum_{k>=1} 1/(k*(k+1))^3 = Sum_{k>=1} 1/A060459(k).

%F Equals 6 * Sum_{k>=2} 1/(k*(k+1)^2*(k+2)) = Sum_{k>=3} 1/A008911(k).

%F Equals 2 * Integral_{x=0..1, y=0..1} x*(1-x)*y*(1-y)/(1-x*y)^2 dx dy.

%F Equals 4 * Sum_{m,n>=1} (m-n)^2/(m*n*(m+1)^2*(n+1)^2*(m+2)*(n+2)) (Sitaru, 2023). - _Amiram Eldar_, Aug 18 2023

%e 0.13039559891064138116550900012384886468630059275920...

%t RealDigits[10 - Pi^2, 10, 100][[1]]

%o (PARI) 10 - Pi^2 \\ _Michel Marcus_, Oct 29 2021

%Y Cf. A002388, A008911, A010467, A060459.

%K nonn,cons

%O 0,2

%A _Amiram Eldar_, Oct 29 2021