A068996 Decimal expansion of 1 - 1/e.

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

Offset: 0

Author

N. J. A. Sloane, Apr 08 2002

Keywords

• nonn
• cons
• easy

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

Examples

			0.6321205588285576784044762...
		

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.
• Brady Haran and Tom Crawford, A Problem with Infinite Hotel Keys, YouTube Numberphile video, 2025.
• 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).
• Index entries for transcendental numbers

Crossrefs

Cf. A000166, A068985, A091131, A185393, A232744.

Programs

Maple

evalf[140](1-exp(-1));  # Alois P. Heinz, May 16 2025

Mathematica

RealDigits[1 - 1/E, 10, 100][[1]] (* Alonso del Arte, Jul 06 2012 *)

PARI

1 - exp(-1) \\ Michel Marcus, Mar 04 2016

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

Equals A091131/e. - Hugo Pfoertner, Aug 20 2024