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 A068996 Decimal expansion of 1 - 1/e. 14
 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, 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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. 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 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 Also equals the mode of a Gompertz distribution when the shape parameter is less than 1. - Jean-François Alcover, Apr 17 2013 The asymptotic density of numbers with an even number of trailing zeros in their factorial base representation (A232744). - Amiram Eldar, Feb 26 2021 REFERENCES Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3, pp. 12-17. Anders Hald, A History of Probability and Statistics and Their Applications before 1750, Wiley, NY, 1990 (Chapter 19). John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65. LINKS Brian Conrey and Tom Davis, Derangements. MathOverflow, What is the effect of adding 1/2 to a continued fraction?. Jonathan Sondow and Eric Weisstein, e, MathWorld. Bala Subramanian, Why time constant is 63.2% not a 50 or 70%? (2018). FORMULA Equals Integral_{x = 0 .. 1} exp(-x) dx. - Alonso del Arte, Jul 06 2012 Equals -Sum_{k>=1} (-1)^k/k!. - Bruno Berselli, May 13 2013 Equals Sum_{k>=0} ((1/((2*k+2)*(2*k)!). - Fred Daniel Kline, Mar 03 2016 From Peter Bala, Nov 27: 2019: (Start) 1 - 1/e = Sum_{n >= 0} n!/(A(n)*A(n+1)), where A(n) = A000522(n). Continued fraction expansion: [0; 1, 1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, ...]. Related continued fraction expansions include 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, ...]; (1/2)*(1 - 1/e) = [0; 3, 6, 10, 14, 18, ..., 4*n + 2, ...]; 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, ...]; (1/4)*(1 - 1/e) = [0; 6, 3, 20, 7, 36, 11, 52, 15, ..., 16*n + 4, 4*n + 3, ...]. (End) Equals Integral_{x=0..1} x * cosh(x) dx. - Amiram Eldar, Aug 14 2020 EXAMPLE 0.6321205588285576784044762... MATHEMATICA RealDigits[1 - 1/E, 10, 100][[1]] (* Alonso del Arte, Jul 06 2012 *) PROG (PARI) 1 - exp(-1) \\ Michel Marcus, Mar 04 2016 CROSSREFS Cf. A000166, A068985, A185393, A232744. Sequence in context: A126445 A277435 A033326 * A068924 A106224 A254571 Adjacent sequences: A068993 A068994 A068995 * A068997 A068998 A068999 KEYWORD nonn,cons,easy AUTHOR N. J. A. Sloane, Apr 08 2002 STATUS approved

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Last modified March 25 17:06 EDT 2023. Contains 361528 sequences. (Running on oeis4.)