A332892 -id:A332892 - cp's OEIS frontend

A143819 Decimal expansion of Sum_{k>=0} 1/(3*k)!.

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

Offset: 1

Peter Bala, Sep 03 2008

Previous name was: Decimal expansion of the constant 1 + 1/3! + 1/6! + 1/9! + ... = 1.16805 83133 75918 ... .
Define a sequence R(n) of real numbers by R(n) := Sum_{k>=0} (3*k)^n/(3*k)! for n = 0,1,2,... . This constant is R(0); the decimal expansions of R(2) - R(1) = 1/1! + 1/4! + 1/7! and R(1) = 1/2! + 1/5! + 1/8! + ... may be found in A143820 and A143821. It is easy to verify that the sequence R(n) satisfies the recurrence relation u(n+3) = 3*u(n+2) - 2*u(n+1) + Sum_{i=0..n} binomial(n,i) * 3^(n-i)*u(i). Hence R(n) is an integral linear combination of R(0), R(1) and R(2) and so also an integral linear combination of R(0), R(1) and R(2) - R(1). Some examples are given below.
Bowman and Mc Laughlin (Corollary 10 with m = -1) give a continued fraction expansion for this constant and deduce the constant is irrational. - Peter Bala, Apr 17 2017

			1.168058313375918525516256929611144747716933295113292516385891232685...
R(n) as a linear combination of R(0), R(1) and R(2) - R(1).
=======================================
  R(n)  |     R(0)     R(1)   R(2)-R(1)
=======================================
  R(3)  |       1        1        3
  R(4)  |       6        2        7
  R(5)  |      25       11       16
  R(6)  |      91       66       46
  R(7)  |     322      352      203
  R(8)  |    1232     1730     1178
  R(9)  |    5672     8233     7242
  R(10) |   32202    39987    43786
  ...
The column entries are from A143815, A143816 and A143817.

D. Bowman and J. Mc Laughlin, Polynomial continued fractions, arXiv:1812.08251 [math.NT], 2018; Acta Arith. 103 (2002), no. 4, 329-342.
Michael Penn, Two sum identities, YouTube video, 2020.
Michael I. Shamos, A catalog of the real numbers, (2011). See p. 209.

Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143820, A143821.

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), this sequence (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.

RealDigits[ N[ 1/3*(2*Cos[Sqrt[3]/2]/Sqrt[E] + E), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)
With[{nn=120},RealDigits[N[Total[Table[1/(3n)!,{n,nn}]]+1,nn],10,nn][[1]]] (* Harvey P. Dale, Apr 20 2013 *)

suminf(k=0, 1/(3*k)!) \\ Michel Marcus, Feb 21 2016

Equals (exp(1) + exp(w) + exp(w^2))/3, where w = exp(2*Pi*i/3).
A143819 + A143820 + A143821 = exp(1).
Equals 1/3 * (e + 2 * cos(sqrt(3)/2) / sqrt(e)). - Bernard Schott, Mar 01 2020
Sum_{k>=0} (-1)^k / (3*k)! = (exp(-1) + 2*exp(1/2)*cos(sqrt(3)/2))/ 3 = 0.83471946857721... - Vaclav Kotesovec, Mar 02 2020
Continued fraction: 1 + 1/(6 - 6/(121 - 120/(505 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (3*n )*(3*n - 1)*(3*n - 2) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

Offset corrected by R. J. Mathar, Feb 05 2009

New name from Bernard Schott, Mar 02 2020

A332890 Decimal expansion of Sum_{k>=0} 1/(4*k)!.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Mar 01 2020

For q integer >= 1, Sum_{m>=0} 1/(q*m)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity.

			1.0416914703416917479394211141000191431669197664918929...

Serge Francinou, Hervé Gianella, Serge Nicolas, Exercices de Mathématiques, Oraux X-ENS, Analyse 2, problème 3.10 p. 182, Cassini, Paris, 2004.

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 76.

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), this sequence (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.

Maple
```
evalf(1/2 * (cos(1) + cosh(1)), 100);
```

RealDigits[Sum[1/(4n)!,{n,0,\[Infinity]}],10,120][[1]] (* Harvey P. Dale, Apr 18 2023 *)

suminf(k=0,(1 + (-1)^k)/((2*k)!))/2 \\ Hugo Pfoertner, Mar 01 2020

suminf(k=0, 1/(4*k)!) \\ Michel Marcus, Mar 02 2020

Equals (1/2) * (cos(1) + cosh(1)).
Equals (1/2) * Sum_{k>=0} (1 + (-1)^k)/((2*k)!). - Peter Luschny, Mar 01 2020
Sum_{k>=0} (-1)^k / (4*k)! = cos(1/sqrt(2)) * cosh(1/sqrt(2)) = 0.958358132833... - Vaclav Kotesovec, Mar 02 2020
Continued fraction: 1 + 1/(24 - 24/(1681 - 1680/(11881 - ... - 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 22 2024

More terms from Hugo Pfoertner, Mar 02 2020

A195390 a(n) = (6*n)!.

Original entry on oeis.org

1, 720, 479001600, 6402373705728000, 620448401733239439360000, 265252859812191058636308480000000, 371993326789901217467999448150835200000000, 1405006117752879898543142606244511569936384000000000, 12413915592536072670862289047373375038521486354677760000000000

Offset: 0

Vincenzo Librandi, Sep 24 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..75

Cf. A000142, A008588, A010050, A100732, A100733, A100734, A271946, A332892.

Magma
```
[Factorial(6*n): n in [0..10]];
```

(6*Range[0,10])! (* Harvey P. Dale, Dec 16 2013 *)

From Amiram Eldar, Apr 03 2021: (Start)
a(n) = A000142(A008588(n)).
Sum_{n>=0} 1/a(n) = A332892. (End)

A269296 Decimal expansion of Sum_{k>=0} 1/(5k)!.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Feb 21 2016

From Vaclav Kotesovec, Feb 24 2016: (Start)
Sum_{k>=0} 1/k! = A001113 = exp(1).
Sum_{k>=0} 1/(2k)! = A073743 = cosh(1).
Sum_{k>=0} 1/(3k)! = A143819 = (2*cos(sqrt(3)/2)*exp(-1/2) + exp(1))/3.
Sum_{k>=0} 1/(4k)! = (cos(1) + cosh(1))/2 = 1.0416914703416917479394211141...
(End)
For q integer >= 1, Sum_{m>=0} 1/(q*m)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity. - Bernard Schott, Mar 02 2020
Continued fraction: 1 + 1/(120 - 120/(30241 - 30240/(360361 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (5*n)*(5*n - 1)*(5*n - 2)*(5*n - 3)*(5*n - 4) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

			1 + 1/5! + 1/10! + 1/15! + ... = 1.008333608907290289976453667354838786...

Eric Weisstein's World of Mathematics, Factorial Sums

Cf. A100734.

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), this sequence (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.

evalf((exp(1) + 2*exp(-(sqrt(5) + 1)/4) * cos(sqrt((5 - sqrt(5))/2)/2) + 2*exp((sqrt(5) - 1)/4) * cos(sqrt((5 + sqrt(5))/2)/2))/5, 120); # Vaclav Kotesovec, Feb 24 2016

RealDigits[HypergeometricPFQ[{}, {1/5, 2/5, 3/5, 4/5}, 1/3125], 10, 120][[1]]

suminf(k=0, 1/(5*k)!) \\ Michel Marcus, Feb 21 2016

Equals Sum_{k>=0} 1/A100734(k).
Equals (exp(1) + exp(-(-1)^(1/5)) + exp((-1)^(2/5)) + exp(-(-1)^(3/5)) + exp((-1)^(4/5)))/5.
Equals (exp(1) + 2*exp(-(sqrt(5) + 1)/4) * cos(sqrt((5 - sqrt(5))/2)/2) + 2*exp((sqrt(5) - 1)/4) * cos(sqrt((5 + sqrt(5))/2)/2))/5. - Vaclav Kotesovec, Feb 24 2016
Sum_{k>=0} (-1)^k / (5*k)! = (exp(-1) + 2*cos(5^(1/4)/(2*sqrt(phi))) * exp(phi/2) + 2*cos(5^(1/4)*sqrt(phi)/2) / exp(1/(2*phi)))/5 = 0.99166694223909419..., where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Mar 02 2020

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A143819 Decimal expansion of Sum_{k>=0} 1/(3*k)!.

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A332890 Decimal expansion of Sum_{k>=0} 1/(4*k)!.

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A195390 a(n) = (6*n)!.

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A269296 Decimal expansion of Sum_{k>=0} 1/(5k)!.

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