OFFSET
0,2
COMMENTS
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.
REFERENCES
Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013, p. 220.
A. M. Mathai, An introduction to geometrical probability: distributional aspects with applications, Amsterdam: Gordon and Breach, 1999, p. 275, ex. 2.5.3.
LINKS
R. W. Gosper, Acceleration of Series, AIM-304 (1974), page 71.
Olivier Schneegans, How Close to 10 is Pi^2?, The American Mathematical Monthly, Vol. 126, No. 5 (2019), p. 448.
Daniel Sitaru, Problem B131, Crux Mathematicorum, Vol. 49, No. 7 (2023), p. 381.
FORMULA
Equals Sum_{k>=1} 1/(k*(k+1))^3 = Sum_{k>=1} 1/A060459(k).
Equals 6 * Sum_{k>=2} 1/(k*(k+1)^2*(k+2)) = Sum_{k>=3} 1/A008911(k).
Equals 2 * Integral_{x=0..1, y=0..1} x*(1-x)*y*(1-y)/(1-x*y)^2 dx dy.
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
EXAMPLE
0.13039559891064138116550900012384886468630059275920...
MATHEMATICA
RealDigits[10 - Pi^2, 10, 100][[1]]
PROG
(PARI) 10 - Pi^2 \\ Michel Marcus, Oct 29 2021
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Oct 29 2021
STATUS
approved