A346441 -id:A346441 - cp's OEIS frontend

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

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

A143821 Decimal expansion of the constant 1/2! + 1/5! + 1/8! + ... = 0.50835 81599 84216 ... .

Original entry on oeis.org

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

Offset: 0

Peter Bala, Sep 03 2008

Define a sequence of real numbers R(n) by R(n) := Sum_{k >= 0} (3*k)^n/(3*k)! for n = 0,1,2... . This constant is R(1); the decimal expansions of R(0) = 1 + 1/3!+ 1/6! + 1/9! + ... and R(2) - R(1) = 1/1! + 1/4! + 1/7! + ... may be found in A143819 and A143820. 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.

			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.

Michael I. Shamos, A catalog of the real numbers, (2011). See p. 460.

Cf. A073742, A073743, A143815, A143816, A143817, A143818, A143819, A143820, A346441.

RealDigits[ N[ -((Cos[Sqrt[3]/2] - E^(3/2) + Sqrt[3]*Sin[Sqrt[3]/2])/(3*Sqrt[E])), 105]][[1]] (* Jean-François Alcover, Nov 08 2012 *)

Constant = (exp(1) + w*exp(w) + w^2*exp(w^2))/3, where w = exp(2*Pi*i/3). A143819 + A143820 + A143821 = exp(1).

Continued fraction: 1/(2 - 2/(61 - 60/(337 - 336/(991 - ... - 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

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

A332892 Decimal expansion of Sum_{k>=0} 1/(6*k)!.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Mar 02 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.

			1.001388890976564743867770084409737409278656173555781142...

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

Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!), A143819 (Sum 1/(3k)!), A332890 (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), this sequence (Sum 1/(6k)!), A346441.

Maple
```
evalf(sum(1/(6*n)!,n=0..infinity),150);
```

RealDigits[(1/3)*(Cosh[1] + 2*Cosh[1/2]*Cos[Sqrt[3]/2]), 10, 120][[1]] (* Amiram Eldar, May 31 2023 *)

sumpos(k=0, 1/(6*k)!) \\ Michel Marcus, Mar 02 2020

Equals (1/3) * (cosh(1) + 2*cosh(1/2)*cos((sqrt(3))/2)).
Sum_{k>=0} (-1)^k / (6*k)! = (cos(1) + 2*cos(1/2)*cosh(sqrt(3)/2))/3 = 0.9986111131987866537... - Vaclav Kotesovec, Mar 02 2020
Continued fraction: 1 + 1/(720 - 720/(665281 - 665280/(13366081 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (6*n)*(6*n - 1)*(6*n - 2)*(6*n - 3)*(6*n - 4)*(6*n - 5) for n >= 1. Cf. A346441. - Peter Bala, Feb 22 2024

A348597 a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k / (3*k)!.

Original entry on oeis.org

1, 1, 2, 5, 20, 100, 601, 4207, 33656, 302903, 3029030, 33319330, 399831961, 5197815493, 72769416902, 1091541253529, 17464660056464, 296899220959888, 5344185977277985, 101539533568281715, 2030790671365634300, 42646604098678320299, 938225290170923046578

Offset: 0

Ilya Gutkovskiy, Mar 25 2022

nonn

Cf. A000166, A009102, A346441, A349087, A352659.

Table[n! Sum[(-1)^k/(3 k)!, {k, 0, Floor[n/3]}], {n, 0, 22}]
nmax = 22; CoefficientList[Series[(Exp[-x] + 2 Exp[x/2] Cos[Sqrt[3] x/2])/(3 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!

E.g.f.: (exp(-x) + 2 * exp(x/2) * cos(sqrt(3)*x/2)) / (3*(1 - x)).

a(n) = round(c * n!), where c = 0.834719468... = A346441.

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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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A143821 Decimal expansion of the constant 1/2! + 1/5! + 1/8! + ... = 0.50835 81599 84216 ... .

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

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A332892 Decimal expansion of Sum_{k>=0} 1/(6*k)!.

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A348597 a(n) = n! * Sum_{k=0..floor(n/3)} (-1)^k / (3*k)!.

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