A249455 - OEIS

%I #15 Feb 08 2024 01:52:00

%S 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,

%T 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,

%U 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

%N 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.

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

%H Steven R. Finch, <a href="/A249455/a249455.pdf">Convex Lattice Polygons</a>, Dec 18 2003. [Cached copy, with permission of the author]

%H Matthew Perkins and Robert A. Van Gorder, <a href="https://doi.org/10.1007/s11139-018-0037-4">Closed-form calculation of infinite products of Glaisher-type related to Dirichlet series</a>, The Ramanujan Journal, Vol. 49 (2019), pp. 371-389; <a href="https://ora.ox.ac.uk/objects/uuid:6f2a85a6-60c7-4d6c-8299-b6ef3ccafad1">alternative link</a>. See Corollary 4.3, p. 386.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

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

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

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

%F 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

%e 1.21306131942526684720759906998236090688383627...

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

%o (PARI) 2/exp(.5) \\ _Charles R Greathouse IV_, Oct 02 2022

%Y Cf. A019774, A092605, A249456.

%Y Cf. A074962, A243262, A243263.

%K nonn,cons,easy

%O 1,2

%A _Jean-François Alcover_, Oct 29 2014