A220608 -id:A220608 - 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

A020077 a(n) = floor( Gamma(n+3/7)/Gamma(3/7) ).

Original entry on oeis.org

1, 0, 0, 1, 5, 22, 122, 787, 5852, 49330, 465112, 4850460, 55433829, 688963306, 9251792972, 133490155743, 2059562402902, 33835668047686, 589707357402530, 10867464157846629, 211139303638163096, 4313274345751046120

Offset: 0

Simon Plouffe

nonn

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

Cf. A220608.

Cf. A020074, A020075, A020076, A020078, A020079.

[Floor(Gamma(n+3/7)/Gamma(3/7)): 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(3/7,n)), n = 0..25); # G. C. Greubel, Nov 17 2019

Floor[Pochhammer[3/7, Range[0, 25]]] (* G. C. Greubel, Nov 17 2019 *)

vector(26, n, my(x=3/7); gamma(n-1+x)\gamma(x) ) \\ G. C. Greubel, Nov 17 2019

[floor(rising_factorial(3/7, n)) for n in (0..25)] # G. C. Greubel, Nov 17 2019

A220609 Decimal expansion of Gamma(4/7).

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Dec 17 2012

See the second comment of A220086.

			1.5585810329024750082750092912459739225208504720945386...

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

Mathematica
```
RealDigits[Gamma[4/7], 10, 90][[1]]
```
Maxima
```
fpprec:90; ev(bfloat(gamma(4/7)));
```

Equals Pi*sec(Pi/14)/A220608.

A020122 Ceiling of Gamma(n+3/7)/Gamma(3/7).

Original entry on oeis.org

1, 1, 1, 2, 6, 23, 123, 788, 5853, 49331, 465113, 4850461, 55433830, 688963307, 9251792973, 133490155744, 2059562402903, 33835668047687, 589707357402531, 10867464157846630, 211139303638163097, 4313274345751046121

Offset: 0

Simon Plouffe

nonn

Robert Israel, Table of n, a(n) for n = 0..449

Cf. A220608.

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

A020032 Nearest integer to Gamma(n + 3/7)/Gamma(3/7).

Original entry on oeis.org

1, 0, 1, 1, 5, 23, 123, 788, 5853, 49330, 465113, 4850460, 55433829, 688963306, 9251792972, 133490155744, 2059562402903, 33835668047686, 589707357402530, 10867464157846630, 211139303638163097, 4313274345751046120

Offset: 0

Simon Plouffe

nonn

Gamma(n + 3/7)/Gamma(3/7) = 1, 3/7, 30/49, 510/343, 12240/2401, 379440/16807, 14418720/117649, 648842400/823543, ... - R. J. Mathar, Sep 04 2016

Cf. A144739, A020077, A020122, A220608.

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

cp's OEIS Frontend

A220086 Decimal expansion of Gamma(1/7).

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A020077 a(n) = floor( Gamma(n+3/7)/Gamma(3/7) ).

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A220609 Decimal expansion of Gamma(4/7).

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A020122 Ceiling of Gamma(n+3/7)/Gamma(3/7).

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Maple

A020032 Nearest integer to Gamma(n + 3/7)/Gamma(3/7).

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Maple