A068985 -id:A068985 - cp's OEIS frontend

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

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

A135002 Decimal expansion of 2/e.

Original entry on oeis.org

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

Offset: 0

Omar E. Pol, Nov 15 2007

From Johannes W. Meijer, Jun 27 2016: (Start)
This constant is related to the values of zeta(2*n-1) of the Riemann zeta function and the Euler Mascheroni constant gamma. If we define Z(n) = (1/n) * (sum(zeta(2*n-2*k-1) * Z(k), k=0..n-2) + gamma * Z(n-1)), with Z(0) = 1, then limit(Z(n), n -> infinity) = 2/exp(1).
Similar formulas appear in A090998 and A112302.
The structure of the n! * Z(n) formulas leads to the multinomial coefficients A036039. (End).

			0.735758882342... = 2*A068985.

Index entries for transcendental numbers

Cf. A036039, A068985, A112302, A090998.

evalf(2/exp(1)) ; # R. J. Mathar, Jul 14 2013

RealDigits[2/E,10,120][[1]] (* Harvey P. Dale, Dec 25 2013 *)

2*exp(-1) \\ Charles R Greathouse IV, Apr 16 2015

Integral of log x from x = 1/e to e. - Charles R Greathouse IV, Apr 16 2015
Equals lim_{k->0} 2*(1 - k)^(1/k). - Ilya Gutkovskiy, Jun 27 2016
Equals Sum_{i>=0} ((-1)^i)(1-i)/i!. - Maciej Kaniewski, Sep 10 2017
Equals Sum_{i>=0} ((-1)^i)(i^2+2)/i!. - Maciej Kaniewski, Sep 12 2017
From Peter Bala, Mar 21 2022: (Start)
2/e = Integral_{x = 1..oo} (2*x/(1+x))^n*(x^2+x+1-n)/x^2*exp(-x) dx;
2/e = - Integral_{x = 0..1} (2*x/(1+x))^n*(x^2+x+1-n)/x^2*exp(-x) dx, both valid for n >= 2. (End)

A137204 Decimal expansion of e + 1/e.

Original entry on oeis.org

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

Offset: 1

Mohammad K. Azarian, Mar 04 2008

			3.086161269630487556955811241514123365203058224731727...

Index entries for transcendental numbers.

Cf. A001113, A068985, A073743.

RealDigits[E + 1/E, 10, 120][[1]] (* Amiram Eldar, Jun 20 2023 *)

exp(1)+1/exp(1) \\ Michel Marcus, Mar 14 2013

Equals 2*A073743. - Bruno Berselli, Mar 14 2013

A174548 Decimal expansion of e - 1/e.

Original entry on oeis.org

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

Offset: 1

Paul Curtz, Mar 22 2010

			2.3504023872876029137647637...

Index entries for transcendental numbers

Cf. A001113 (e), A068985 (1/e), and A137204 (e + 1/e), A073742 (sinh(1)).

evalf(exp(1)-exp(-1)) ; # R. J. Mathar, Oct 14 2011

Mathematica
```
RealDigits[E - 1/E, 10, 111][[1]]
```

exp(1) - exp(-1) \\ Michel Marcus, May 05 2019

Equals 2 * sinh(1) = 2 * A073742. - Amiram Eldar, Nov 25 2020

Edited and extended by Robert G. Wilson v, Apr 25 2010

A032634 a(n) = floor(n/e).

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27

Offset: 0

Patrick De Geest, May 15 1998

Differs from A057367 first at a(49). - R. J. Mathar, Jan 28 2013

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Beatty sequences

Beatty seq. of A068985.

A032634 := proc(n)
        floor(n/exp(1)) ;
end proc: # R. J. Mathar, Jan 28 2013

Array[Floor[#/E] &, 75, 0] (* Michael De Vlieger, Oct 06 2019 *)

A059193 Engel expansion of 1/e = 0.367879... .

Original entry on oeis.org

3, 10, 28, 54, 88, 130, 180, 238, 304, 378, 460, 550, 648, 754, 868, 990, 1120, 1258, 1404, 1558, 1720, 1890, 2068, 2254, 2448, 2650, 2860, 3078, 3304, 3538, 3780, 4030, 4288, 4554, 4828, 5110, 5400, 5698, 6004, 6318, 6640, 6970, 7308, 7654, 8008, 8370, 8740

Offset: 1

Mitch Harris

Cf. A006784 for definition of Engel expansion.

Friedrich Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191.

G. C. Greubel and T. D. Noe, Table of n, a(n) for n = 1..1000[Terms 1 to 300 computed by T. D. Noe; Terms 301 to 1000 computed by G. C. Greubel, Dec 27 2016]
Friedrich Engel, Entwicklung der Zahlen nach Stammbruechen, Verhandlungen der 52. Versammlung deutscher Philologen und Schulmaenner in Marburg, 1913, pp. 190-191. English translation by Georg Fischer, included with his permission.
Paul Erdős and Jeffrey Shallit, New bounds on the length of finite Pierce and Engel series, Sem. Theor. Nombres Bordeaux (2) 3 (1991), no. 1, 43-53.
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.
Eric Weisstein's World of Mathematics, Engel Expansion.
Index entries for sequences related to Engel expansions.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A006784, A068985.

EngelExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@
NestList[{Ceiling[1/Expand[#[[1]] #[[2]] - 1]], Expand[#[[1]] #[[2]] - 1]/1} &, {Ceiling[1/(A - Floor[A])], (A - Floor[A])/1}, n - 1]];
EngelExp[N[1/E, 7!], 100] (* Modified by G. C. Greubel, Dec 27 2016 *)
Join[{3}, Table[2*(2*n+1)*(n-1), {n, 1, 200}]] (* Vladimir Joseph Stephan Orlovsky, Jun 26 2011 *)
Join[{3},LinearRecurrence[{3,-3,1},{10,28,54},50]] (* Harvey P. Dale, May 10 2012 *)

Vec(x*(3 + x + 7*x^2 - 3*x^3)/(1-x)^3 + O(x^50)) \\ G. C. Greubel, Dec 27 2016

a(n) = 2*(2*n+1)*(n-1) (for n>1) follows from 1/e = Sum_{n>=1} (1/(2*n)! - 1/(2*n+1)!). - Helena Verrill (verrill(AT)math.lsu.edu), Jan 19 2004
a(1)=3, a(2)=10, a(1)=28, a(2)=54, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Harvey P. Dale, May 10 2012
From G. C. Greubel, Dec 27 2016: (Start)
G.f.: x*(3 + x + 7*x^2 - 3*x^3)/(1-x)^3.
E.g.f.: 2 + 3*x + 2*(2*x^2 + x - 1)*exp(x). (End)
From Amiram Eldar, May 05 2025: (Start)
Sum_{n>=1} 1/a(n) = 7/9 - log(2)/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = 1/9 + Pi/12 - log(2)/6. (End)

A242674 Decimal expansion of the asymptotic probability of success in one of the Secretary problems.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, May 20 2014

This is the asymptotic probability of success for the full-information problem with uniform distribution: we can not only determine which of any two applicants is better than the other, but also determine his/her absolute value, and that value is known to be uniformly distributed on a known interval (say, [0, 1]), independently for each applicant; so we have more information than in the basic version of the problem (for which the chance of success is given by A068985), so the chance of success is greater. Here the number of applicants is known in advance (although we consider the limiting case when it is sent to infinity); for the variant where it is itself a random variable, see A325905. - Andrey Zabolotskiy, Sep 14 2019

			0.580164223920855346426008323572997276633...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.15, p. 362.

Eric Weisstein's MathWorld, Sultan's Dowry Problem
Wikipedia, Secretary problem

Cf. A054404, A242672, A242673, A068985, A325905.

a = x /. FindRoot[ExpIntegralEi[x] - EulerGamma - Log[x] == 1, {x, 1}, WorkingPrecision -> 105]; Exp[-a] - (Exp[a]-a-1)*ExpIntegralEi[-a] // RealDigits[#, 10, 100]& // First

exp(-a) - (exp(a)-a-1)*Ei(-a), where a is the unique real solution of the equation Ei(a)-gamma-log(a) = 1, Ei being the exponential integral function, and gamma the Euler-Mascheroni constant (0.5772156649...).

A052520 Number of pairs of sequences of cardinality at least 2.

Original entry on oeis.org

0, 0, 0, 0, 24, 240, 2160, 20160, 201600, 2177280, 25401600, 319334400, 4311014400, 62270208000, 958961203200, 15692092416000, 271996268544000, 4979623993344000, 96035605585920000, 1946321606541312000, 41359334139002880000, 919636959090769920000, 21356013827774545920000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Vincenzo Librandi, Table of n, a(n) for n = 0..400
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 87.

Cf. sequences with formula (n + k)*n! listed in A282466.

Cf. A001113, A001620, A068985, A091725, A099285.

Concatenation([0,0,0,0], List([4..20], n-> (n-3)*Factorial(n))); # G. C. Greubel, May 13 2019

[n le 3 select 0 else (n-3)*Factorial(n): n in [0..20]]; // G. C. Greubel, May 13 2019

spec := [S,{B=Sequence(Z,2 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Table[Sum[n!, {i,4,n}], {n, 0, 19}] (* Zerinvary Lajos, Jul 12 2009 *)
With[{nn=20},CoefficientList[Series[x^4/(x-1)^2,{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 03 2016 *)

{a(n) = if(n<4, 0, (n-3)*n!)}; \\ G. C. Greubel, May 13 2019

[0,0,0,0]+[(n-3)*factorial(n) for n in (4..20)] # G. C. Greubel, May 13 2019

E.g.f.: x^4/(1-x)^2.
(n-3)*a(n+1) + (2+n-n^2)*a(n) = 0, with a(0) = a(1) = a(2) = a(3) = 0, a(4) = 24.
a(n) = (n-3)*n!, n>2.
a(n) = (n+1)!*(n-3)/(n+1), n>2. - Gary Detlefs, Oct 02 2011
From Amiram Eldar, Jan 14 2021: (Start)
Sum_{n>=4} 1/a(n) = 59/36 - 2*e/3 - gamma/6 + Ei(1)/6 = 59/36 - (2/3)*A001113 - (1/6)*A001620 + A091725/2.
Sum_{n>=4} (-1)^n/a(n) = 1/36 - 1/(3*e) + gamma/6 - Ei(-1)/6 = 1/36 - (1/3)*A068985 + (1/6)*A001620 + (1/6)*A099285. (End)

cp's OEIS Frontend

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

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

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

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A135002 Decimal expansion of 2/e.

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A137204 Decimal expansion of e + 1/e.

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A174548 Decimal expansion of e - 1/e.

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A032634 a(n) = floor(n/e).

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A059193 Engel expansion of 1/e = 0.367879... .