A144749 -id:A144749 - cp's OEIS frontend

A178394 Decimal expansion of e!.

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

A144713 Decimal expansion of phi/(phi^phi - 1), where phi is the golden ratio (1+sqrt(5))/2.

Original entry on oeis.org

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

Offset: 1

Daniel Akaiya (pi216n(AT)hotmail.com), Sep 19 2008

Contains the golden ratio, the unit one and a tetration of the golden ratio.

			1.373009968938968060437092447225959813351682960081209926...

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

Cf. A001622, A144749.

RealDigits[GoldenRatio/(GoldenRatio^GoldenRatio-1),10,120][[1]] (* Harvey P. Dale, Dec 16 2017 *)

numerical_approx(golden_ratio/(golden_ratio^golden_ratio - 1), digits=120) # G. C. Greubel, Jun 16 2022

More terms from N. J. A. Sloane, Sep 19 2008

A092134 Continued fraction expansion for phi^phi, where phi is the golden ratio (1+sqrt(5))/2.

Original entry on oeis.org

2, 5, 1, 1, 1, 1, 10, 1, 1, 2, 8, 7643, 4, 1, 51, 2, 2, 8, 5, 2, 1, 6, 5, 4, 1, 42, 2, 1, 1, 1, 1, 1, 1, 1, 6, 2, 6, 2, 12, 2, 1, 6, 3, 13, 11, 2, 9, 2, 1, 4, 1, 2, 1, 6, 3, 1, 1, 1, 11, 3, 1, 2, 1, 1, 2, 3, 3, 1, 2, 3, 1, 56, 1, 24, 6, 20, 3, 27, 2

Offset: 0

Ryan Witko (witko(AT)nyu.edu), Mar 30 2004

			2.178457.. = 2 + 1/(5 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...))))).

Cf. A144749 (decimal expansion).

with(numtheory):Digits:= 300: x:=(sqrt(5)+1)/2:convert(evalf(x^x), confrac); # Michel Lagneau

phi := (1 + Sqrt[5])/2; ContinuedFraction[phi^phi, 80] (* Bruno Berselli, Aug 07 2013 *)

gr = (1+sqrt(5))/2; contfrac(gr^gr) \\ Michel Marcus, Aug 07 2013

a(12) and following terms corrected by Charles R Greathouse IV, Aug 07 2013

Offset changed by Andrew Howroyd, Aug 07 2024

A181779 Duplicate of A092134.

Original entry on oeis.org

2, 5, 1, 1, 1, 1, 10, 1, 1, 2, 8, 7643, 4, 1, 51, 2, 2, 8, 5, 2, 1, 6, 5, 4, 1, 42, 2, 1, 1, 1, 1, 1, 1, 1, 6, 2, 6, 2, 12, 2, 1, 6, 3, 13, 11, 2, 9, 2, 1, 4, 1, 2, 1, 6, 3, 1, 1, 1, 11, 3, 1, 2, 1, 1, 2, 3, 3, 1, 2, 3, 1, 56, 1, 24, 6, 20, 3, 27, 2, 1, 2, 1, 2, 5, 2, 1, 1, 14, 1, 91, 1, 2, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 36, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 16, 21

Offset: 0

Michel Lagneau

dead

Previous name was: Continued fraction for phi^phi.

			2.178457567937599147372545... = 2 + 1/(5 + 1/(1 + 1/(1 + 1/(1 + 1/(1 + ...))))).

H. Walser, The Golden Section, Math. Assoc. of Amer, Washington DC 2001.
C. J. Willard, Le nombre d'or, Magnard, Paris 1987.

Cf. A144749 (decimal expansion).

with(numtheory):Digits:= 300: x:=(sqrt(5)+1)/2:convert(evalf(x^x), confrac);

ContinuedFraction[GoldenRatio^ GoldenRatio, 100 ]

phi=(1+sqrt(5))/2;contfrac(phi^phi) \\ Charles R Greathouse IV, Jul 29 2011

Offset changed and missing term inserted by Andrew Howroyd, Jul 08 2024

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A178394 Decimal expansion of e!.

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A144713 Decimal expansion of phi/(phi^phi - 1), where phi is the golden ratio (1+sqrt(5))/2.

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A092134 Continued fraction expansion for phi^phi, where phi is the golden ratio (1+sqrt(5))/2.

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A181779 Duplicate of A092134.

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