cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

A256778 Decimal expansion of the generalized Euler constant gamma(1,4).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 10 2015

Keywords

Examples

			0.71028979306409369731376647579508261030406104249693294...
		

References

  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.3, p. 32.

Crossrefs

Cf. A001620 (EulerGamma), A016627, A020777, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256779-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

Programs

  • Magma
    R:=RealField(100); (2*EulerGamma(R) + Pi(R) + 2*Log(2))/8; // G. C. Greubel, Aug 27 2018
  • Mathematica
    RealDigits[EulerGamma/4 + Pi/8 + Log[2]/4, 10, 103] // First
  • PARI
    default(realprecision, 100); (2*Euler + Pi + 2*log(2))/8 \\ G. C. Greubel, Aug 27 2018
    

Formula

Equals (2*EulerGamma + Pi + 2*log(2))/8.
Equals Sum_{n>=0} (1/(4n+1) - 1/2*arctanh(2/(4n+3))).
Equals -(psi(1/4) + log(4))/4 = (A020777 - A016627)/4. - Amiram Eldar, Jan 07 2024

A256779 Decimal expansion of the generalized Euler constant gamma(1,5).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 10 2015

Keywords

Examples

			0.735920396831617584189289725844752893059997383987625...
		

Crossrefs

Cf. A001620 (EulerGamma), A016628, A200135, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/5 + Pi(R)/10*Sqrt(1 + 2/Sqrt(5)) + Log(5)/20 + Sqrt(5)/10*Log((1 + Sqrt(5))/2); // G. C. Greubel, Aug 28 2018
  • Mathematica
    RealDigits[-Log[5]/5 - PolyGamma[1/5]/5, 10, 105] // First
  • PARI
    Euler/5 + Pi/10*sqrt(1 + 2/sqrt(5)) + log(5)/20 + sqrt(5)/10*log((1 + sqrt(5))/2) \\ Michel Marcus, Apr 10 2015
    

Formula

Equals EulerGamma/5 + Pi/10*sqrt(1 + 2/sqrt(5)) + log(5)/20 + sqrt(5)/10*log((1 + sqrt(5))/2).
Equals Sum_{n>=0} (1/(5n+1) - 2/5*arctanh(5/(10n+7))).
Equals -(psi(1/5) + log(5))/5 = (A200135 - A016628)/5. - Amiram Eldar, Jan 07 2024

A256843 Decimal expansion of the generalized Euler constant gamma(2,3).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 11 2015

Keywords

Examples

			0.07320737570615959366903185990752913907462383026830934562939...
		

Crossrefs

Cf. A001620 (gamma(1,1) = EulerGamma), A002391, A200064.
Primitive ruler-and-compass constructible gamma(r,k): A228725 (1,2), A256425 (1,3), A256778 (1,4), A256779 (1,5), A256780 (2,5), A256781 (1,8), A256782 (3,8), A256783 (1,12), A256784 (5,12).
Other gamma(r,k) (1 <= r <= k <= 5): A239097 (2,2), A256843 (2,3), A256844 (3,3), A256845 (2,4), A256846 (3,4), A256847 (4,4), A256848 (3,5), A256849 (4,5), A256850 (5,5).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/3 - Pi(R)/(6*Sqrt(3)) + Log(3)/6; // G. C. Greubel, Aug 28 2018
  • Mathematica
    Join[{0}, RealDigits[-Log[3]/3 - PolyGamma[2/3]/3, 10, 105] // First]
  • PARI
    default(realprecision, 100); Euler/3 - Pi/(6*sqrt(3)) + log(3)/6 \\ G. C. Greubel, Aug 28 2018
    

Formula

Equals EulerGamma/3 - Pi/(6*sqrt(3)) + log(3)/6.
Equals -(psi(2/3) + log(3))/3 = (A200064 - A002391)/3. - Amiram Eldar, Jan 07 2024

A256780 Decimal expansion of the generalized Euler constant gamma(2,5).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 10 2015

Keywords

Examples

			0.190389326430203154225983229764268163260151948448458487...
		

Crossrefs

Cf. A001620 (EulerGamma), A016628, A200136, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/5 + Pi(R)/10*Sqrt(1 - 2/Sqrt(5)) + Log(5)/20 - Sqrt(5)/10*Log((1 + Sqrt(5))/2); // G. C. Greubel, Aug 28 2018
  • Mathematica
    RealDigits[-Log[5]/5 - PolyGamma[2/5]/5, 10, 105] // First
  • PARI
    Euler/5 + Pi/10*sqrt(1 - 2/sqrt(5)) + log(5)/20 - sqrt(5)/10*log((1 + sqrt(5))/2) \\ Michel Marcus, Apr 10 2015
    

Formula

Equals EulerGamma/5 + Pi/10*sqrt(1 - 2/sqrt(5)) + log(5)/20 - sqrt(5)/10*log((1 + sqrt(5))/2).
Equals Sum_{n>=0} (1/(5n+2) - 2/5*arctanh(5/(10n+9))).
Equals -(psi(2/5) + log(5))/5 = (A200136 - A016628)/5. - Amiram Eldar, Jan 07 2024

A256781 Decimal expansion of the generalized Euler constant gamma(1,8).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 10 2015

Keywords

Examples

			0.788631390202002367443880819838976661978118204921...
		

Crossrefs

Cf. A001620 (EulerGamma), A016631, A228725 (gamma(1,2)), A250129, A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/8 + (1/8)*(Pi(R)/2*(Sqrt(2)+1) + Log(2) + Sqrt(2)*Log(Sqrt(2) + 1)); // G. C. Greubel, Aug 28 2018
  • Mathematica
    RealDigits[-3/8*Log[2] - PolyGamma[1/8]/8, 10, 105] // First
  • PARI
    Euler/8 + 1/8*(Pi/2*(sqrt(2)+1) + log(2) + sqrt(2)*log(sqrt(2) + 1)) \\ Michel Marcus, Apr 10 2015
    

Formula

Equals EulerGamma/8 + 1/8*(Pi/2*(sqrt(2)+1) + log(2) + sqrt(2)*log(sqrt(2) + 1)).
Equals Sum_{n>=0} (1/(8n+1) - 1/4*arctanh(4/(8n+5))).
Equals -(psi(1/8) + log(8))/8 = -(A250129 + A016631)/8. - Amiram Eldar, Jan 07 2024

A256782 Decimal expansion of the generalized Euler constant gamma(3,8).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 10 2015

Keywords

Examples

			0.08431968843316295593904035680375480012812437382591706852303...
		

Crossrefs

Cf. A001620 (EulerGamma), A016631, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)), A354633.

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/8 + (1/8)*(Pi(R)/2*(Sqrt(2)-1) + Log(2) - Sqrt(2)*Log(Sqrt(2)+1)); // G. C. Greubel, Aug 28 2018
  • Mathematica
    Join[{0}, RealDigits[-3/8*Log[2] - PolyGamma[3/8]/8, 10, 104] // First]
  • PARI
    default(realprecision, 100); Euler/8 + 1/8*(Pi/2*(sqrt(2)-1) + log(2) - sqrt(2)*log(sqrt(2)+1)) \\ G. C. Greubel, Aug 28 2018
    

Formula

Equals EulerGamma/8 + 1/8*(Pi/2*(sqrt(2)-1) + log(2) - sqrt(2)*log(sqrt(2)+1)).
Equals -(psi(3/8) + log(8))/8 = -(A354633 + A016631)/8. - Amiram Eldar, Jan 07 2024

A256845 Decimal expansion of the generalized Euler constant gamma(2,4).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 11 2015

Keywords

Examples

			0.1443039162253832151516280225206006077605398339849808997...
		

Crossrefs

Cf. A001620 (gamma(1,1) = EulerGamma),
Primitive ruler-and-compass constructible gamma(r,k): A228725 (1,2), A256425 (1,3), A256778 (1,4), A256779 (1,5), A256780 (2,5), A256781 (1,8), A256782 (3,8), A256783 (1,12), A256784 (5,12),
Other gamma(r,k) (1 <= r <= k <= 5): A239097 (2,2), A256843 (2,3), A256844 (3,3), A256845 (2,4), A256846 (3,4), A256847 (4,4), A256848 (3,5), A256849 (4,5), A256850 (5,5).

Programs

  • Magma
    R:= RealField(100); EulerGamma(R)/4; // G. C. Greubel, Aug 28 2018
  • Mathematica
    RealDigits[EulerGamma/4, 10, 107] // First
  • PARI
    default(realprecision, 100); Euler/4 \\ G. C. Greubel, Aug 28 2018
    

Formula

-log(4)/4 - PolyGamma(1/2)/4 = EulerGamma/4
From Amiram Eldar, Jul 21 2020: (Start)
Equals -Integral_{x=0..oo} e^(-x^2)*x*log(x) dx.
Equals Integral_{x=0..oo} (e^(-x^4) - e^(-x^2))/x dx. (End)

A256846 Decimal expansion of the generalized Euler constant gamma(3,4) (negated).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 11 2015

Keywords

Examples

			-0.07510837033335461230189437002479311074523130734684351439...
		

Crossrefs

Cf. A001620 (gamma(1,1) = EulerGamma),
Primitive ruler-and-compass constructible gamma(r,k): A228725 (1,2), A256425 (1,3), A256778 (1,4), A256779 (1,5), A256780 (2,5), A256781 (1,8), A256782 (3,8), A256783 (1,12), A256784 (5,12),
Other gamma(r,k) (1 <= r <= k <= 5): A239097 (2,2), A256843 (2,3), A256844 (3,3), A256845 (2,4), A256846 (3,4), A256847 (4,4), A256848 (3,5), A256849 (4,5), A256850 (5,5).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/4 - Pi(R)/8 - Log(4)/4 + Log(8)/4; // G. C. Greubel, Aug 28 2018
  • Mathematica
    Join[{0}, RealDigits[-Log[4]/4 - PolyGamma[3/4]/4, 10, 104] // First ]
  • PARI
    default(realprecision, 100); Euler/4 - Pi/8 - log(4)/4 + log(8)/4 \\ G. C. Greubel, Aug 28 2018
    

Formula

-log(4)/4 - PolyGamma(3/4)/4 = EulerGamma/4 - Pi/8 - log(4)/4 + log(8)/4

A256848 Decimal expansion of the generalized Euler constant gamma(3,5) (negated).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 11 2015

Keywords

Examples

			-0.013763739708181991968019076883991139603013419915782102729...
		

Crossrefs

Cf. A001620 (gamma(1,1) = EulerGamma),
Primitive ruler-and-compass constructible gamma(r,k): A228725 (1,2), A256425 (1,3), A256778 (1,4), A256779 (1,5), A256780 (2,5), A256781 (1,8), A256782 (3,8), A256783 (1,12), A256784 (5,12),
Other gamma(r,k) (1 <= r <= k <= 5): A239097 (2,2), A256843 (2,3), A256844 (3,3), A256845 (2,4), A256846 (3,4), A256847 (4,4), A256848 (3,5), A256849 (4,5), A256850 (5,5).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/5 + Pi(R)/(10*Sqrt(2*(5+Sqrt(5)))) - Pi(R)/(2*Sqrt(10*(5+Sqrt(5)))) + Log(5)/20 + Log((5-Sqrt(5))/(5+Sqrt(5)))/(4*Sqrt(5)); // G. C. Greubel, Aug 28 2018
  • Mathematica
    Join[{0}, RealDigits[-Log[5]/5 - PolyGamma[3/5]/5, 10, 108] // First  ]
  • PARI
    default(realprecision, 100); Euler/5 + Pi/(10*sqrt(2*(5+sqrt(5)))) - Pi/(2*sqrt(10*(5+sqrt(5)))) + log(5)/20 + log((5-sqrt(5))/(5+sqrt(5)))/(4*sqrt(5)) \\ G. C. Greubel, Aug 28 2018
    

Formula

Equals -log(5)/5 - PolyGamma(3/5)/5.
Equals EulerGamma/5 + Pi/(10*sqrt(2*(5+sqrt(5)))) - Pi/(2*sqrt(10*(5+sqrt(5)))) + log(5)/20 + log((5-sqrt(5))/(5+sqrt(5)))/(4*sqrt(5)).

A256849 Decimal expansion of the generalized Euler constant gamma(4,5) (negated).

Original entry on oeis.org

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

Views

Author

Jean-François Alcover, Apr 11 2015

Keywords

Examples

			-0.12888586914559238304189234001387044397828817291465897856 ...
		

Crossrefs

Cf. A001620 (gamma(1,1) = EulerGamma),
Primitive ruler-and-compass constructible gamma(r,k): A228725 (1,2), A256425 (1,3), A256778 (1,4), A256779 (1,5), A256780 (2,5), A256781 (1,8), A256782 (3,8), A256783 (1,12), A256784 (5,12),
Other gamma(r,k) (1 <= r <= k <= 5): A239097 (2,2), A256843 (2,3), A256844 (3,3), A256845 (2,4), A256846 (3,4), A256847 (4,4), A256848 (3,5), A256849 (4,5), A256850 (5,5).

Programs

  • Magma
    SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/5 - Pi(R)/(10*Sqrt(2*(5-Sqrt(5)))) - Pi(R)/(2*Sqrt(10*(5-Sqrt(5)))) + Log(5)/20 - Log(5-Sqrt(5))/(4*Sqrt(5)) + Log(5+Sqrt(5))/( 4*Sqrt(5)); // G. C. Greubel, Aug 28 2018
  • Mathematica
    RealDigits[-Log[5]/5 - PolyGamma[4/5]/5, 10, 107] // First
  • PARI
    default(realprecision, 100); Euler/5 - Pi/(10*sqrt(2*(5-sqrt(5)))) - Pi/(2*sqrt(10*(5-sqrt(5)))) + log(5)/20 - log(5-sqrt(5))/(4*sqrt(5)) + log(5+sqrt(5))/(4*sqrt(5)) \\ G. C. Greubel, Aug 28 2018
    

Formula

Equals -log(5)/5 - PolyGamma(4/5)/5.
Equals EulerGamma/5 - Pi/(10*sqrt(2*(5-sqrt(5)))) - Pi/(2*sqrt(10*(5-sqrt(5)))) + log(5)/20 - log(5-sqrt(5))/(4*sqrt(5)) + log(5+sqrt(5))/(4*sqrt(5)).
Showing 1-10 of 14 results. Next