A111293 -id:A111293 - cp's OEIS frontend

A255196 The continued fraction expansion of Pi! (A111293).

7, 5, 3, 6, 2, 1, 1, 3, 1, 3, 1, 1, 1, 1, 6, 1, 7, 22, 22, 1, 6, 2, 1, 24, 3, 2, 3, 1, 1, 13, 1, 19, 1, 42, 2, 3, 1, 4, 1, 1, 7, 4, 2, 4, 5, 1, 2, 5, 3, 7, 3, 1, 1, 7, 2, 1, 1, 1, 11, 5, 3, 4, 48, 1, 5, 2, 1, 5, 3, 429, 1, 4, 8, 1, 5, 2, 6, 1, 5, 1

Offset: 0

Robert G. Wilson v, Feb 17 2015

Records: 5, 6, 22, 24, 42, 48, 429, 2962, 136067, 259575, ..., .

			Pi! = 7.188082728976032702082194345124758718559301763968437162410035699423438403...
= 7 + 1/(5 + 1/(3 + 1/(6 + 1/(2 + ...))))
= [a_0; a_1, a_2, a_3, ...] = [7; 5, 3, 6, 2, ...]

Cf. A111293.

Mathematica
```
ContinuedFraction[Pi!, 100]
```

default(realprecision, 100); contfrac(gamma(Pi+1)) \\ Michel Marcus, Feb 18 2015

A178394 Decimal expansion of e!.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 27 2010

Gamma(e+1).

			4.2608204763570033817001212246457024649334243739593219749116...

Alois P. Heinz, Table of n, a(n) for n = 1..1000
D. M. Bătinetu-Giurgiu, Problem 11808, The American Mathematical Monthly, Vol. 121, No. 10 (2014), p. 947; A Gamma Integral Limit, Solution to Problem 11808 by Roberto Tauraso, ibid., Vol. 123, No. 9 (2016), pp. 944-945.

Cf. A073233, A111293, A144749.

Mathematica
```
RealDigits[N[E!,200]]
```

Equals Integral_{x>=0} x^e/e^x dx. - Alois P. Heinz, Aug 27 2015

Equals lim_{n->oo} n^2 * Integral_{x=n!^(1/n)..(n+1)!^(1/(n+1))} Gamma(n*x) dx (Bătinetu-Giurgiu, 2014). - Amiram Eldar, Mar 26 2022

A178840 Decimal expansion of the factorial of Golden Ratio.

Original entry on oeis.org

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

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jun 17 2010

			1.44922960226989660037787979062976833708408989096667607533702385813891...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A111293, A178394, A178839.

Cf. A001622 (golden ratio).

SetDefaultRealField(RealField(100)); Gamma((Sqrt(5)-1)/2); // G. C. Greubel, Jan 21 2019

evalf(GAMMA(1+evalf((1+sqrt(5))/2,100)),106); # Golden ratio

RealDigits[Gamma[(Sqrt[5] - 1)/2], 10, 120][[1]] (* Vaclav Kotesovec, Jan 20 2019 *)

default(realprecision, 100); gamma((sqrt(5)-1)/2) \\ G. C. Greubel, Jan 21 2019

numerical_approx(gamma(1/golden_ratio), digits=100) # G. C. Greubel, Jan 21 2019

Factorial of Golden Ratio = Gamma(1 + phi) = Gamma((3 + sqrt(5))/2). - Bernard Schott, Jan 21 2019

Equals Gamma((sqrt(5) - 1)/2). - Vaclav Kotesovec, Jan 21 2019

A178839 Decimal expansion of the factorial of Euler-Mascheroni constant.

Original entry on oeis.org

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

Offset: 0

Paolo P. Lava and Giorgio Balzarotti, Jun 17 2010

			0.891152726131463856163339689050466491784711126227363757289227682281...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A111293, A178394, A178840.

SetDefaultRealField(RealField(100)); R:=RealField(); Gamma(EulerGamma(R) + 1); // G. C. Greubel, Jan 21 2019

evalf(GAMMA(1+evalf(gamma,100)),106); # Euler-Mascheroni

RealDigits[Gamma[1 + EulerGamma], 10, 120][[1]] (* Vaclav Kotesovec, Jan 20 2019 *)

default(realprecision, 100); gamma(1 + Euler) \\ G. C. Greubel, Jan 21 2019

numerical_approx(gamma(1 + euler_gamma), digits=100) # G. C. Greubel, Jan 21 2019

Offset corrected by Vaclav Kotesovec, Jan 20 2019

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A255196 The continued fraction expansion of Pi! (A111293).

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A178394 Decimal expansion of e!.

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A178840 Decimal expansion of the factorial of Golden Ratio.

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A178839 Decimal expansion of the factorial of Euler-Mascheroni constant.

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Maple

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