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A068996 Decimal expansion of 1 - 1/e. 7

%I

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

%T 3,2,5,5,4,1,8,8,8,6,8,9,6,8,2,3,2,1,6,5,4,9,2,1,6,3,1,9,8,3,0,2,5,3,

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

%N Decimal expansion of 1 - 1/e.

%C From the "derangements" problem: this is the probability that if a large number of people are given their hats at random, at least one person gets their own hat.

%C 1-1/e is the limit to which (1 - !n/n!) {= 1 - A000166(n)/A000142(n) = A002467(n)/A000142(n)} converges as n tends to infinity. - _Lekraj Beedassy_, Apr 14 2005

%C Also, this is lim_{n->inf} P(n), where P(n) is the probability that a random rooted forest on [n] is not connected. - _Washington Bomfim_, Nov 01 2010

%C Also equals the mode of a Gompertz distribution when the shape parameter is less than 1. - _Jean-Fran├žois Alcover_, Apr 17 2013

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.

%D A. Hald, A History of Probability and Statistics and Their Applications before 1750, Wiley, NY, 1990 (Chapter 19).

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.

%H B. Conrey & T. Davis, <a href="http://www.geometer.org/mathcircles/derange.pdf">Derangements</a>

%H J. Sondow and Eric Weisstein, <a href="http://mathworld.wolfram.com/e.html">e</a>, MathWorld

%H Bala Subramanian, <a href="https://electronics.stackexchange.com/questions/396653/why-time-constant-is-63-2-not-a-50-or-70">Why time constant is 63.2% not a 50 or 70%?</a> (2018)

%F integral(x = 0 .. 1, exp(-x) dx ). - _Alonso del Arte_, Jul 06 2012

%F Equals -sum(k>=1, (-1)^k/k! ). - _Bruno Berselli_, May 13 2013

%F Also equals Sum_{k>=0} ((1/((2*k+2)*(2*k)!). - _Fred Daniel Kline_, Mar 03 2016

%e 0.6321205588285576784044762...

%t RealDigits[1 - 1/E, 10, 100][[1]] (* _Alonso del Arte_, Jul 06 2012 *)

%o (PARI) 1 - exp(-1) \\ _Michel Marcus_, Mar 04 2016

%Y Cf. A000166, A068985, A185393.

%K nonn,cons,easy

%O 0,1

%A _N. J. A. Sloane_, Apr 08 2002

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Last modified November 14 04:44 EST 2019. Contains 329108 sequences. (Running on oeis4.)