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
Examples
1.21306131942526684720759906998236090688383627...
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.
- Index entries for transcendental numbers.
Programs
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Mathematica
RealDigits[2/Sqrt[E], 10, 100] // First
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PARI
2/exp(.5) \\ Charles R Greathouse IV, Oct 02 2022
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