A348670 Decimal expansion of 10 - Pi^2.

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, 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, 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

Offset: 0

Author

Amiram Eldar, Oct 29 2021

Keywords

• nonn
• cons

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.

Examples

			0.13039559891064138116550900012384886468630059275920...
		

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.
• Index entries for transcendental numbers.

Crossrefs

Cf. A002388, A008911, A010467, A060459.

Programs

Mathematica

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

PARI

10 - Pi^2 \\ Michel Marcus, Oct 29 2021

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