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

0,2

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

Table of n, a(n) for n=0..104.

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.

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

Cf. A002388, A008911, A010467, A060459.

Sequence in context: A132330 A117078 A021333 * A104141 A279977 A060533

Adjacent sequences: A348667 A348668 A348669 * A348671 A348672 A348673

Keyword

nonn,cons

Author

Amiram Eldar, Oct 29 2021

Status

approved