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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.
6
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
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.
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
KEYWORD
nonn,cons,easy
AUTHOR
STATUS
approved