A220609 -id:A220609 - cp's OEIS frontend

A220086 Decimal expansion of Gamma(1/7).

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

Offset: 1

Bruno Berselli, Dec 12 2012

(A220086/A220605)*(A220607/A220606) = A160389, which is the case n=7 of (Gamma(1/n)/Gamma(2/n))*(Gamma((n-1)/n)/Gamma((n-2)/n)) = 2*cos(Pi/n).
A220086*A220605*A220606*A220607*A220608*A220609 = (2*Pi)^3/sqrt(7), which is the case n=7 of product(Gamma(i/n), i=1..n-1) = sqrt((2*Pi)^(n-1)/n) (see also the second link to Wikipedia).
Continued fraction expansion: 6, 1, 1, 4, 1, 2, 2, 1, 5, 1, 10, 7, 1,...

			6.5480629402478244377140933494289962626211351873841351...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Wikipedia, Particular values of the Gamma function: General rational arguments
Wikipedia, Particular values of the Gamma function: Products
Index to sequences related to the Gamma function

SetDefaultRealField(RealField(100)); Gamma(1/7); // G. C. Greubel, Mar 10 2018

Mathematica
```
RealDigits[Gamma[1/7], 10, 90][[1]]
```
Maxima
```
fpprec:90; ev(bfloat(gamma(1/7)));
```

default(realprecision, 100); gamma(1/7) \\ G. C. Greubel, Mar 10 2018

Equals Pi * csc(Pi/7) / A220607, where csc is the cosecant function.
(A220086/A220605) * (A220607/A220606) = A160389, which is the case n=7 of (Gamma(1/n)/Gamma(2/n))*(Gamma((n-1)/n)/Gamma((n-2)/n)) = 2*cos(Pi/n).
A220086*A220605*A220606*A220607*A220608*A220609 = (2*Pi)^3/sqrt(7), which is the case n=7 of product(Gamma(i/n), i=1..n-1) = sqrt((2*Pi)^(n-1)/n) (see also the second link to Wikipedia).
A220086*A220607 = A220605*A220606 + A220608*A220609. - Andrea Pinos, May 21 2025

A020076 a(n) = floor( Gamma(n+4/7)/Gamma(4/7) ).

Original entry on oeis.org

1, 0, 0, 2, 8, 37, 210, 1380, 10450, 89574, 857351, 9063434, 104876887, 1318452295, 17893281147, 260730668150, 4059948975490, 67279154450985, 1182190856781603, 21954973054515489, 429690186924088870

Offset: 0

Simon Plouffe

nonn

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A220609.

Cf. A020074, A020075, A020077, A020078, A020079.

[Floor(Gamma(n+4/7)/Gamma(4/7)): n in [0..25]]; // G. C. Greubel, Nov 17 2019

Digits := 64:f := proc(n,x) trunc(GAMMA(n+x)/GAMMA(x)); end;
seq(floor(pochhammer(4/7,n)), n = 0..25); # G. C. Greubel, Nov 17 2019

Floor[Pochhammer[4/7, Range[0, 25]]] (* G. C. Greubel, Nov 17 2019 *)

vector(26, n, my(x=4/7); gamma(n-1+x)\gamma(x) ) \\ G. C. Greubel, Nov 17 2019

[floor(rising_factorial(4/7, n)) for n in (0..25)] # G. C. Greubel, Nov 17 2019

A220608 Decimal expansion of Gamma(3/7).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Dec 17 2012

See the second comment of A220086.

			2.0675117265602293530246124063088269435592142114923875...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index to sequences related to the Gamma function

Mathematica
```
RealDigits[Gamma[3/7], 10, 90][[1]]
```
Maxima
```
fpprec:90; ev(bfloat(gamma(3/7)));
```

Equals Pi*sec(Pi/14)/A220609.

A020121 Ceiling of GAMMA(n+4/7)/GAMMA(4/7).

Original entry on oeis.org

1, 1, 1, 3, 9, 38, 211, 1381, 10451, 89575, 857352, 9063435, 104876888, 1318452296, 17893281148, 260730668151, 4059948975491, 67279154450986, 1182190856781604, 21954973054515490, 429690186924088871

Offset: 0

Simon Plouffe

nonn

Cf. A220609.

Digits := 64:f := proc(n,x) ceil(GAMMA(n+x)/GAMMA(x)); end;

A020031 Nearest integer to Gamma(n + 4/7)/Gamma(4/7).

Original entry on oeis.org

1, 1, 1, 2, 8, 38, 210, 1380, 10450, 89574, 857352, 9063435, 104876887, 1318452295, 17893281148, 260730668151, 4059948975491, 67279154450986, 1182190856781603, 21954973054515490, 429690186924088871

Offset: 0

Simon Plouffe

nonn

a(n) equals A020121(n) or A020076(n). - R. J. Mathar, May 18 2007

Cf. A020121, A020076, A220609.

Digits := 64:f := proc(n,x) round(GAMMA(n+x)/GAMMA(x)); end;

Round[Gamma[Range[0,20]+4/7]/Gamma[4/7]] (* Harvey P. Dale, Dec 24 2023 *)

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A220086 Decimal expansion of Gamma(1/7).

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A020076 a(n) = floor( Gamma(n+4/7)/Gamma(4/7) ).

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A220608 Decimal expansion of Gamma(3/7).

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A020121 Ceiling of GAMMA(n+4/7)/GAMMA(4/7).

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Maple

A020031 Nearest integer to Gamma(n + 4/7)/Gamma(4/7).

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Maple

Mathematica