A396820 Decimal expansion of (479 - 52*sqrt(2))/576.

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, 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, 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

Offset

0,1

Comments

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.

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

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).

References

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

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

Links

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

Robert Deltheil, Sur la théorie des probabilités géométriques, Annales de la faculté des sciences de Toulouse, 3rd series, Vol. 11 (1919), pp. 1-65. See p. 56.

Robert Deltheil, Probabilités Géométriques, Gauthier-Villars, Paris, 1926, p. 54.

Maurice G. Kendall and P. A. P. Moran, Geometrical Probability, Charles Griffin and Co. Ltd., 1963, p. 46.

Herbert Solomon, Geometric Probability, Philadelphia: Society for Industrial and Applied Mathematics, 1978, p. 114.

Eric Weisstein's World of Mathematics, Sylvester's Four-Point Problem.

Index entries for algebraic numbers, degree 2.

Formula

Equals 1 - 4*A396661.

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).

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

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

Example

0.703925164507984474760958656842180034573571289028469...

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A220359 A021591 A384475 * A316599 A077185 A385809

Adjacent sequences: A396817 A396818 A396819 * A396821 A396822 A396823

Keyword

nonn,cons

Author

Amiram Eldar, Jun 07 2026

Status

approved