A030652 -id:A030652 - cp's OEIS frontend

A073006 Decimal expansion of Gamma(2/3).

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

This constant is transcendental: Chudnovsky famously proved that Gamma(1/3) is algebraically independent of Pi, but Gamma(1/3)*Gamma(2/3) = 2*Pi/sqrt(3) by the reflection formula. - Charles R Greathouse IV, Aug 21 2023

			1.354117939426400416945288028154513785519327266056793698394022467963782...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:8 at page 413.

Harry J. Smith, Table of n, a(n) for n = 1..5000
Simon Plouffe, GAMMA(2/3).
Index to sequences related to the Gamma function.
Index entries for transcendental numbers.

Cf. A030652 (continued fraction). - Harry J. Smith, May 14 2009

Cf. A073005, A186706.

SetDefaultRealField(RealField(100)); Gamma(2/3); // G. C. Greubel, Mar 10 2018

Mathematica
```
RealDigits[ N[ Gamma[2/3], 110]][[1]]
```

allocatemem(932245000); default(realprecision, 5080); x=gamma(2/3); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b073006.txt", n, " ", d));  \\ Harry J. Smith, May 14 2009

Gamma(2/3) * A073005 = A186706. - R. J. Mathar, Jun 18 2006

A030651 Continued fraction for Gamma(1/3).

Original entry on oeis.org

2, 1, 2, 8, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 9, 1, 4, 1, 3, 1, 7, 1, 1, 3, 6, 2, 2, 1, 1, 32, 3, 3, 3, 1, 24, 2, 2, 25, 1, 2, 1, 6, 2, 1, 1, 3, 1, 9, 3, 2, 1, 6, 7, 2, 8, 2, 5, 1, 5, 1, 2, 2, 2, 2, 4, 3, 1, 5, 1, 15, 1, 1, 2, 4, 3, 3, 1, 5, 1, 4, 1, 8, 1, 3, 1, 1, 8, 2, 1, 2, 1, 514, 1, 2, 1, 1, 1

Offset: 0

Paolo Dominici (pl.dm(AT)libero.it)

			Gamma(1/3) = 2.67893853470774... = 2 + 1/(1 + 1/(2 + 1/(8 + 1/(1 + ...)))). - _Harry J. Smith_, Apr 20 2009

Harry J. Smith, Table of n, a(n) for n = 0..999
Gang Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A030652, A073005 (decimal expansion).

ContinuedFraction[Gamma[1/3], 50] (* Alonso del Arte, Mar 30 2020 *)

{ default(realprecision, 1080); x=contfrac(gamma(1/3)); for (n=1, 1000, write("b030651.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009

Note that 3 * Gamma(1/3) * Gamma(2/3) = 2 * Pi * sqrt(3).

Offset changed by Andrew Howroyd, Aug 03 2024

A322509 Factorial expansion of Gamma(2/3) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

1, 0, 2, 0, 2, 2, 6, 6, 0, 3, 1, 11, 7, 6, 6, 14, 1, 8, 12, 15, 8, 17, 8, 1, 13, 15, 3, 4, 10, 16, 25, 1, 25, 22, 6, 3, 19, 17, 8, 10, 25, 37, 29, 17, 35, 19, 24, 25, 30, 31, 4, 7, 51, 49, 14, 51, 45, 54, 0, 26, 34, 41, 56, 57, 16, 15, 63, 4, 51, 42, 13, 35, 12, 15, 66, 22, 13, 43, 14, 78

Offset: 1

G. C. Greubel, Dec 12 2018

nonn

			Gamma(2/3) = 1 + 0/2! + 2/3! + 0/4! + 2/5! + 2/6! + 6/7! + 6/8! + ...

Index entries for factorial base representation

Cf. A073006 (decimal expansion), A030652 (continued fraction).

Cf. A068463 (Gamma(3/4)), A068464 (Gamma(1/4)), A322508 (Gamma(1/3)).

SetDefaultRealField(RealField(250));  [Floor(Gamma(2/3))] cat [Floor(Factorial(n)*Gamma(2/3)) - n*Floor(Factorial((n-1))*Gamma(2/3)) : n in [2..80]];

With[{b = Gamma[2/3]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]]

default(realprecision, 250); b = gamma(2/3); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", "))

b=gamma(2/3);
def a(n):
    if (n==1): return floor(b)
    else: return expand(floor(factorial(n)*b) -n*floor(factorial(n-1)*b))
[a(n) for n in (1..80)]

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A073006 Decimal expansion of Gamma(2/3).

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A030651 Continued fraction for Gamma(1/3).

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A322509 Factorial expansion of Gamma(2/3) = Sum_{n>=1} a(n)/n!.

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