A030651 -id:A030651 - cp's OEIS frontend

A030652 Continued fraction for Gamma(2/3).

1, 2, 1, 4, 1, 2, 8, 1, 1, 4, 4, 1, 6, 12, 1, 5, 1, 1, 1, 3, 2, 1, 1, 1, 18, 9, 1, 42, 1, 1, 2, 1, 1, 10, 3, 2, 4, 6, 2, 11, 1, 1, 8, 65, 9, 4, 1, 11, 2, 3, 1, 4, 3, 1, 2, 1, 2, 1, 5, 1, 1, 1, 2, 1, 14, 1, 5, 1, 6, 2, 7, 1, 29, 1, 1, 1, 3, 1, 2, 1, 26, 1, 1, 7, 13, 1, 2, 2, 8, 3, 4, 2, 2, 2

Offset: 0

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

			1.354117939426400416945288028... = 1 + 1/(2 + 1/(1 + 1/(4 + 1/(1 + ...)))). - _Harry J. Smith_, May 14 2009

Harry J. Smith, Table of n, a(n) for n = 0..4999
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A030651, A073006 (decimal expansion).

ContinuedFraction[Gamma[2/3],100] (* Harvey P. Dale, Sep 05 2021 *)

{ allocatemem(932245000); default(realprecision, 5200); x=contfrac(gamma(2/3)); for (n=1, 5000, write("b030652.txt", n-1, " ", x[n])); } \\ Harry J. Smith, May 14 2009

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

Offset changed by Andrew Howroyd, Aug 03 2024

A322508 Factorial expansion of Gamma(1/3) = Sum_{n>=1} a(n)/n!.

Original entry on oeis.org

2, 1, 1, 0, 1, 2, 5, 6, 7, 2, 1, 8, 5, 7, 9, 12, 13, 10, 10, 13, 17, 18, 5, 1, 6, 3, 26, 13, 20, 29, 8, 31, 27, 19, 21, 27, 5, 14, 12, 3, 9, 37, 34, 40, 14, 29, 35, 12, 27, 4, 36, 22, 24, 11, 31, 37, 12, 5, 47, 14, 22, 18, 51, 20, 51, 4, 15, 54, 61, 26, 55, 2, 6, 73, 7, 17, 66, 54, 27

Offset: 1

G. C. Greubel, Dec 12 2018

nonn

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

Index entries for factorial base representation

Cf. A073005 (decimal expansion), A030651 (continued fraction).

Cf. A068463 (Gamma(3/4)), A068464 (Gamma(1/4)), A322509 (Gamma(2/3)).

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

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

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

b=gamma(1/3);
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)]

cp's OEIS Frontend

A030652 Continued fraction for Gamma(2/3).

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