A346435 -id:A346435 - cp's OEIS frontend

A068985 Decimal expansion of 1/e.

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

Offset: 0

N. J. A. Sloane, Apr 08 2002

From the "derangements" problem: this is the probability that if a large number of people are given their hats at random, nobody gets their own hat.
Also, decimal expansion of cosh(1)-sinh(1). - Mohammad K. Azarian, Aug 15 2006
Also, this is lim_{n->inf} P(n), where P(n) is the probability that a random rooted forest on [n] is a tree. See linked file. - Washington Bomfim, Nov 01 2010
Also, location of the minimum of x^x. - Stanislav Sykora, May 18 2012
Also, -1/e is the global minimum of x*log(x) at x = 1/e and the global minimum of x*e^x at x = -1. - Rick L. Shepherd, Jan 11 2014
Also, the asymptotic probability of success in the secretary problem (also known as the sultan's dowry problem). - Andrey Zabolotskiy, Sep 14 2019
The asymptotic density of numbers with an odd number of trailing zeros in their factorial base representation (A232745). - Amiram Eldar, Feb 26 2021
For large range size s where numbers are chosen randomly r times, the probability when r = s that a number is randomly chosen exactly 1 time. Also the chance that a number was not chosen at all. The general case for the probability of being chosen n times is (r/s)^n / (n! * e^(r/s)). - Mark Andreas, Oct 25 2022

			1/e = 0.3678794411714423215955237701614608674458111310317678... = A135005/5.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, Sections 1.3 and 5,23,3, pp. 14, 409.
Anders Hald, A History of Probability and Statistics and Their Applications Before 1750, Wiley, NY, 1990 (Chapter 19).
John Harris, Jeffry L. Hirst, and Michael Mossinghoff, Combinatorics and Graph Theory, Springer Science & Business Media, 2009, p. 161.
L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (103) on page 20.
Traian Lalescu, Problem 579, Gazeta Matematică, Vol. 6 (1900-1901), p. 148.
John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.
Manfred R. Schroeder, Number Theory in Science and Communication, Springer Science & Business Media, 2008, ch. 9.5 Derangements.
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 26, page 233.
Walter D. Wallis and John C. George, Introduction to Combinatorics, CRC Press, 2nd ed. 2016, theorem 5.2 (The Derangement Series).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 27.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Washington Bomfim, Probabilities of connected random forests and derangements, Oct 31 2010.
James Grime and Brady Haran, Derangements, Numberphile video, 2017.
Peter J. Larcombe, Jack Sutton, and James Stanton, A note on the constant 1/e, Palest. J. Math. (2023) Vol. 12, No. 2, 609-619.
Gérard P. Michon, Final Answers: Inclusion-Exclusion.
Michael Penn, A cool, quick limit, YouTube video, 2022.
Eric Weisstein's World of Mathematics, Derangement.
Eric Weisstein's World of Mathematics, Factorial Sums.
Eric Weisstein's World of Mathematics, Spherical Bessel Function of the First Kind.
Eric Weisstein's World of Mathematics, Sultan's Dowry Problem.
Eric Weisstein's World of Mathematics, e.
OEIS Wiki, Number of derangements.
Index entries for transcendental numbers.

Cf. A000166, A001113, A068996, A092553, A232745.
Cf. A059193.
Cf. asymptotic probabilities of success for other "nothing but the best" variants of the secretary problem: A325905, A242674, A246665.
Cf. A049470, A346441, A346440, A346439, A346438, A346437, A346436, A346435, A196498.

RealDigits[N[1/E,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2009 *)

default(realprecision, 110);
exp(-1) \\ Rick L. Shepherd, Jan 11 2014

Equals 2*(1/3! + 2/5! + 3/7! + ...). [Jolley]
Equals 1 - Sum_{i >= 1} (-1)^(i - 1)/i!. [Michon]
Equals lim_{x->infinity} (1 - 1/x)^x. - Arkadiusz Wesolowski, Feb 17 2012
Equals j_1(i)/i = cos(i) + i*sin(i), where j_1(z) is the spherical Bessel function of the first kind and i = sqrt(-1). - Stanislav Sykora, Jan 11 2017
Equals Sum_{i>=0} ((-1)^i)/i!. - Maciej Kaniewski, Sep 10 2017
Equals Sum_{i>=0} ((-1)^i)(i^2+1)/i!. - Maciej Kaniewski, Sep 12 2017
From Peter Bala, Oct 23 2019: (Start)
The series representation 1/e = Sum_{k >= 0} (-1)^k/k! is the case n = 0 of the following series acceleration formulas:
1/e = n!*Sum_{k >= 0} (-1)^k/(k!*R(n,k)*R(n,k+1)), n = 0,1,2,..., where R(n,x) = Sum_{k = 0..n} (-1)^k*binomial(n,k)*k!*binomial(-x,k) are the row polynomials of A094816. (End)
1/e = 1 - Sum_{n >= 0} n!/(A(n)*A(n+1)), where A(n) = A000522(n). - Peter Bala, Nov 13 2019
Equals Integral_{x=0..1} x * sinh(x) dx. - Amiram Eldar, Aug 14 2020
Equals lim_{x->oo} (x!)^(1/x)/x. - L. Joris Perrenet, Dec 08 2020
Equals lim_{n->oo} (n+1)!^(1/(n+1)) - n!^(1/n) (Lalescu, 1900-1901). - Amiram Eldar, Mar 29 2022

More terms from Rick L. Shepherd, Jan 11 2014

A049470 Decimal expansion of cos(1).

Original entry on oeis.org

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

Offset: 0

Albert du Toit (dutwa(AT)intekom.co.za), N. J. A. Sloane

Also, decimal expansion of the real part of e^i. - Bruno Berselli, Feb 08 2013

By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 13 2019

			0.5403023058681397...

Muniru A Asiru, Table of n, a(n) for n = 0..2000
Mohammad K. Azarian, Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 36, No. 5, November 2005, p. 413-414.
Mohammad K. Azarian, Solution of Forty-Five Nested Equilateral Triangles and cosecant of 1 degree, Problem 813, College Mathematics Journal, Vol. 37, No. 5, November 2006, pp. 394-395.
I. S. Gradsteyn and I. M. Ryzhik, Table of integrals, series and products, (1980), page 10 (formula 0.245.7).
Simon Plouffe, cos(1)
Eric Weisstein's World of Mathematics, Factorial Sums
Index entries for transcendental numbers

Cf. A049469 (imaginary part of e^i), A211883 (real part of -(i^e)), A211884 (imaginary part of -(i^e)). - Bruno Berselli, Feb 08 2013
Cf. A073743 ( cosh(1) ), A073448, A275651.
Cf. A068985, A346441, A346440, A346439, A346438, A346437, A346436, A346435, A196498.

evalf(cos(1)); # Altug Alkan, Sep 22 2018

Mathematica
```
RealDigits[Cos[1], 10, 110] [[1]]
```

cos(1) \\ Charles R Greathouse IV, Jan 04 2016

Continued fraction representation: cos(1) = 1/(1 + 1/(1 + 2/(11 + 12/(29 + ... + (2*n - 2)*(2*n - 3)/((4*n^2 - 2*n - 1) + ... ))))). See A275651 for proof. Cf. A073743. - Peter Bala, Sep 02 2016
Equals Sum_{k >= 0} (-1)^k/A010050(k), where A010050(k) = (2k)! [See Gradshteyn and Ryzhik]. - A.H.M. Smeets, Sep 22 2018
Equals 1/A073448. - Alois P. Heinz, Jan 23 2023
From Gerry Martens, May 04 2024: (Start)
Equals (4*(cos(1/4)^4 + sin(1/4)^4) - 3).
Equals (16*(cos(1/4)^6 + sin(1/4)^6) - 10)/6. (End)

A346441 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(3*k)!.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 17 2021

			0.8347194685772109622192832392...

Peter Bala, A continued fraction for A346441
D. Bowman and J. Mc Laughlin, Polynomial continued fractions, Acta Arith. 103 (2002), no. 4, 329-342.
Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A068985, A049470, A143819, A346440, A346439, A346438, A346437, A346436, A346435, A196498.

RealDigits[Sum[(-1)^k/(3*k)!, {k, 0, Infinity}], 10, 100][[1]] (* Amiram Eldar, Jul 18 2021 *)

sumalt(k=0, (-1)^k/(3*k)!) \\ Michel Marcus, Jul 18 2021

Equals 1/(3*e) + 2*sqrt(e)*cos(sqrt(3)/2)/3. - Amiram Eldar, Jul 18 2021

Continued fraction: 1/(1 + 1/(5 + 6/(119 + 120/(503 + ... + P(n-1)/((P(n) - 1) + ... ))))), where P(n) = (3*n)*(3*n - 1)*(3*n - 2) for n >= 1. See Bowman and Mc Laughlin, Corollary 10, p. 341 with m = 1, who also show that the constant is irrational. - Peter Bala, Feb 21 2024

A196498 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(10*k)!.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Oct 03 2011

			0.99999972442680776055212523675802047...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A195070, A195394.

Cf. A068985, A049470, A346441, A346440, A346439, A346438, A346437, A346436, A346435.

RealDigits[ HypergeometricPFQ[{}, {1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10}, -10^-10], 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)

sumalt(k=0, (-1)^k/(10*k)!) \\ Michel Marcus, Jul 18 2021

6 more digits from Jean-François Alcover, Feb 12 2013

A346437 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(7*k)!.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 17 2021

			0.99980158731305804716545837...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A068985, A049470, A346441, A346440, A346439, A346438, A346437, A346436, A346435, A196498.

RealDigits[HypergeometricPFQ[{}, {1/7, 2/7, 3/7, 4/7, 5/7, 6/7}, -1/7^7], 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)

sumalt(k=0, (-1)^k/(7*k)!) \\ Michel Marcus, Jul 18 2021

A346440 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(4*k)!.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 17 2021

			0.95835813283300701621040446...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A068985, A049470, A346441, A346439, A346438, A346437, A346436, A346435, A196498.

RealDigits[Sum[(-1)^k/(4*k)!, {k, 0, Infinity}], 10, 100][[1]] (* Amiram Eldar, Jul 18 2021 *)

sumalt(k=0, (-1)^k/(4*k)!) \\ Michel Marcus, Jul 18 2021

Equals cos(sqrt(2)/2)*cosh(sqrt(2)/2). - Amiram Eldar, Jul 18 2021

Continued fraction: 1/(1 + 1/(23 + 24/(1679 + ... + P(n-1)/((P(n) - 1) + ... )))), where P(n) = (4*n)*(4*n - 1)*(4*n - 2)*(4*n - 3) for n >= 1. Cf. A346441. - Peter Bala, Feb 21 2024

A346436 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(8*k)!.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 17 2021

			0.9999751984127462074717349605281...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A068985, A049470, A346441, A346440, A346439, A346438, A346437, A346435, A196498.

RealDigits[HypergeometricPFQ[{}, {1/8, 1/4, 3/8, 1/2, 5/8, 3/4, 7/8}, -1/2^24], 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)

sumalt(k=0, (-1)^k/(8*k)!) \\ Michel Marcus, Jul 18 2021

A346438 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(6*k)!.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 17 2021

			0.9986111131987866537058529349...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A068985, A049470, A346441, A346440, A346439, A346437, A346436, A346435, A196498.

RealDigits[(Cos[1] + 2*Cos[1/2]*Cosh[Sqrt[3]/2])/3, 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)

sumalt(k=0, (-1)^k/(6*k)!) \\ Michel Marcus, Jul 18 2021

Equals (cos(1) + 2*cos(1/2)*cosh(sqrt(3)/2))/3. - Amiram Eldar, Jun 04 2023

A346439 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(5*k)!.

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 17 2021

			0.9916669422390941905634229...

Michael I. Shamos, A catalog of the real numbers (2011).

Cf. A068985, A049470, A346441, A346440, A346438, A346437, A346436, A346435, A196498.

RealDigits[HypergeometricPFQ[{}, {1/5, 2/5, 3/5, 4/5}, -1/5^5], 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)

sumalt(k=0, (-1)^k/(5*k)!) \\ Michel Marcus, Jul 18 2021

cp's OEIS Frontend

A068985 Decimal expansion of 1/e.

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A049470 Decimal expansion of cos(1).

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A346441 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(3*k)!.

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A196498 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(10*k)!.

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A346437 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(7*k)!.

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A346440 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(4*k)!.

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A346436 Decimal expansion of the constant Sum_{k>=0} (-1)^k/(8*k)!.

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