%I #83 Aug 20 2024 02:38:32

%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-François Alcover_, Apr 17 2013

%C The asymptotic density of numbers with an even number of trailing zeros in their factorial base representation (A232744). - _Amiram Eldar_, Feb 26 2021

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3, pp. 12-17.

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

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

%H Brian Conrey and Tom Davis, <a href="http://www.geometer.org/mathcircles/derange.pdf">Derangements</a>.

%H MathOverflow, <a href="https://mathoverflow.net/questions/128676/what-is-the-effect-of-adding-1-2-to-a-continued-fraction">What is the effect of adding 1/2 to a continued fraction?</a>.

%H Jonathan 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).

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

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

%F From _Peter Bala_, Nov 27 2019: (Start)

%F 1 - 1/e = Sum_{n >= 0} n!/(A(n)*A(n+1)), where A(n) = A000522(n).

%F Continued fraction expansion: [0; 1, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ...].

%F Related continued fraction expansions include

%F 2*(1 - 1/e) = [1; 3, 1, 3, 1, 1, 1, 3, 3, 3, 1, 3, 1, 3, 5, 3, 1, 5, ..., 1, 3, 2*n + 1, 3, 1, 2*n + 1, ...];

%F (1/2)*(1 - 1/e) = [0; 3, 6, 10, 14, 18, ..., 4*n + 2, ...];

%F 4*(1 - 1/e) = [2; 1, 1, 8, 3, 1, 1, 1, 1, 7, 1, 1, 2, 1, 1, 1, 2, 7, 1, 2, 2, 1, 1, 1, 3, ..., 7, 1, n, 2, 1, 1, 1, n+1, ...];

%F (1/4)*(1 - 1/e) = [0; 6, 3, 20, 7, 36, 11, 52, 15, ..., 16*n + 4, 4*n + 3, ...]. (End)

%F Equals Integral_{x=0..1} x * cosh(x) dx. - _Amiram Eldar_, Aug 14 2020

%F Equals A091131/e. - _Hugo Pfoertner_, Aug 20 2024

%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, A091131, A185393, A232744.

%K nonn,cons,easy

%O 0,1

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