A249455 Decimal expansion of 2/sqrt(e), a constant appearing in the expression of the asymptotic expected volume V(d) of the convex hull of randomly selected n(d) vertices (with replacement) of a d-dimensional unit cube.

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

Offset

1,2

References

Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 634.

Links

Table of n, a(n) for n=1..100.

Steven R. Finch, Convex Lattice Polygons, Dec 18 2003. [Cached copy, with permission of the author]

Matthew Perkins and Robert A. Van Gorder, Closed-form calculation of infinite products of Glaisher-type related to Dirichlet series, The Ramanujan Journal, Vol. 49 (2019), pp. 371-389; alternative link. See Corollary 4.3, p. 386.

Index entries for transcendental numbers.

Formula

Lim_{d -> infinity} V(d) =

0 if n(d) <= (2/sqrt(e) - epsilon)^d

1 if n(d) >= (2/sqrt(e) + epsilon)^d.

Equals Product_{m>=1} A(2*m)^((-1)^(m+1)*Pi^(2*m)/(2*m)!), where A(k) is the k-th generalized Glaisher-Kinkelin (or Bendersky-Adamchik) constant (A074962, A243262, A243263, ...) (Perkins and Van Gorder, 2019). - Amiram Eldar, Feb 08 2024

Example

1.21306131942526684720759906998236090688383627...

Mathematica

RealDigits[2/Sqrt[E], 10, 100] // First

Prog

(PARI) 2/exp(.5) \\ Charles R Greathouse IV, Oct 02 2022

Crossrefs

Cf. A019774, A092605, A249456.

Cf. A074962, A243262, A243263.

Sequence in context: A112469 A330502 A368213 * A248967 A331094 A160169

Adjacent sequences: A249452 A249453 A249454 * A249456 A249457 A249458

Keyword

nonn,cons,easy

Author

Jean-François Alcover, Oct 29 2014

Status

approved