

A068996


Decimal expansion of 1  1/e.


8



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
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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.
11/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.  JeanFrançois 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. 1217.
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

Table of n, a(n) for n=0..98.
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,changed


AUTHOR

N. J. A. Sloane, Apr 08 2002


STATUS

approved



