This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A348670 #19 Feb 07 2024 15:33:54
%S A348670 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 A348670 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 A348670 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 A348670 Decimal expansion of 10 - Pi^2.
%C A348670 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 A348670 Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013, p. 220.
%D A348670 A. M. Mathai, An introduction to geometrical probability: distributional aspects with applications, Amsterdam: Gordon and Breach, 1999, p. 275, ex. 2.5.3.
%H A348670 R. W. Gosper, <a href="https://dspace.mit.edu/handle/1721.1/6088">Acceleration of Series</a>, AIM-304 (1974), page 71.
%H A348670 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 A348670 Daniel Sitaru, <a href="https://cms.math.ca/publications/crux/issue/?volume=49&issue=7">Problem B131</a>, Crux Mathematicorum, Vol. 49, No. 7 (2023), p. 381.
%H A348670 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A348670 Equals Sum_{k>=1} 1/(k*(k+1))^3 = Sum_{k>=1} 1/A060459(k).
%F A348670 Equals 6 * Sum_{k>=2} 1/(k*(k+1)^2*(k+2)) = Sum_{k>=3} 1/A008911(k).
%F A348670 Equals 2 * Integral_{x=0..1, y=0..1} x*(1-x)*y*(1-y)/(1-x*y)^2 dx dy.
%F A348670 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 A348670 0.13039559891064138116550900012384886468630059275920...
%t A348670 RealDigits[10 - Pi^2, 10, 100][[1]]
%o A348670 (PARI) 10 - Pi^2 \\ _Michel Marcus_, Oct 29 2021
%Y A348670 Cf. A002388, A008911, A010467, A060459.
%K A348670 nonn,cons
%O A348670 0,2
%A A348670 _Amiram Eldar_, Oct 29 2021