A396820 - OEIS

%I #11 Jun 28 2026 15:50:58

%S 7,0,3,9,2,5,1,6,4,5,0,7,9,8,4,4,7,4,7,6,0,9,5,8,6,5,6,8,4,2,1,8,0,0,

%T 3,4,5,7,3,5,7,1,2,8,9,0,2,8,4,6,9,9,6,5,6,1,5,9,9,4,1,9,0,3,2,0,2,8,

%U 1,0,9,5,6,9,4,3,9,3,1,1,4,5,4,8,2,2,8,9,0,3,1,5,0,9,7,6,8,0,2,4,6,1,4,2,2

%N Decimal expansion of (479 - 52*sqrt(2))/576.

%C The probability that the convex hull of four points independently and uniformly selected at random in the interior of a regular octagon is a quadrilateral.

%C Deltheil (1919, 1926) calculated the wrong value (2851 + 2013*sqrt(2))/(4032 + 2880*sqrt(2)) = 0.7030... .

%C Kendall and Moran (1963) cited this result, and Solomon (1978) gave the wrong complementary probability based on this result (although he also gave the correct general formula based on Alikoski's paper), a value that is cited by Gelca and Andreescu (2017).

%D H. A. Alikoski, Über das Sylvestersche Vierpunktproblem, Ann. Acad. Sci. Fenn., Vol. 51, No. 7 (1939), pp. 1-10.

%D Răzvan Gelca and Titu Andreescu, Putnam and Beyond, 2nd ed., Springer, 2017, p. 843.

%H Robert Deltheil, <a href="https://doi.org/10.5802/afst.311">Sur la théorie des probabilités géométriques</a>, Annales de la faculté des sciences de Toulouse, 3rd series, Vol. 11 (1919), pp. 1-65. See p. 56.

%H Robert Deltheil, <a href="https://archive.org/details/probabilitesgeom0002rdel/page/54/mode/1up">Probabilités Géométriques</a>, Gauthier-Villars, Paris, 1926, p. 54.

%H Maurice G. Kendall and P. A. P. Moran, <a href="https://archive.org/details/geometricalproba033077mbp/page/n49/mode/1up">Geometrical Probability</a>, Charles Griffin and Co. Ltd., 1963, p. 46.

%H Herbert Solomon, <a href="https://archive.org/details/GeometricProbability/page/114/mode/1up">Geometric Probability</a>, Philadelphia: Society for Industrial and Applied Mathematics, 1978, p. 114.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SylvestersFour-PointProblem.html">Sylvester's Four-Point Problem</a>.

%H <a href="/index/Al#algebraic_02">Index entries for algebraic numbers, degree 2</a>.

%F Equals 1 - 4*A396661.

%F Equals 1 - (9*cos(w)^2 + 52*cos(w) + 44)/(9*n^2*sin(w)^2) with w = 2*Pi/n and n = 8 (Alikoski's general formula for a regular n-gon).

%F Equals (2833 + 2031*sqrt(2))/(4032 + 2880*sqrt(2)).

%F Minimal polynomial: 331776*x^2 - 551808*x + 224033.

%e 0.703925164507984474760958656842180034573571289028469...

%t RealDigits[(479 - 52*Sqrt[2])/576, 10, 120][[1]]

%o (PARI) (479 - 52*sqrt(2))/576

%Y Cf. A051050, A051051, A242780 (disk), A394805, A396661, A396819 (regular pentagon), A396821 (regular 10-gon).

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Jun 07 2026