A047787 -id:A047787 - cp's OEIS frontend

A256425 Decimal expansion of the generalized Euler constant gamma(1,3).

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

Offset: 0

N. J. A. Sloane, Apr 09 2015

			0.67780716378423221053372461245491438316931257963255620415268...

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

G. C. Greubel, Table of n, a(n) for n = 0..5000
D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), 125-142. [gamma=A001620, gamma(0,2)=A239097, gamma(1,2)=A228725, N_k=A007947, gamma(1,3)=A256425, the present sequence]

Cf. A001620, A002391, A007947, A047787, A228725, A239097.

RealDigits[EulerGamma/3 + Pi*Sqrt[3]/18 + Log[3]/6, 10, 50][[1]] (* G. C. Greubel, Jun 08 2017 *)

Euler/3 + Pi*sqrt(3)/18 + log(3)/6 \\ Charles R Greathouse IV, Sep 08 2015

-(psi(1/3) + log(3))/3 \\ Charles R Greathouse IV, Feb 03 2025

Equals gamma/3+Pi*sqrt(3)/18+log(3)/6.

Equals -(psi(1/3) + log(3))/3 = (A047787 - A002391)/3. - Amiram Eldar, Jan 07 2024

A097663 Decimal expansion of the constant 3*exp(psi(1/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Aug 25 2004

This constant appears in Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link) and is involved in the exact determination of asymptotic limits of certain order-3 linear recursions with varying coefficients (see A097677 for example).

			0.23311909318456411730537562326544289574460858702592456414096...

A. M. Odlyzko, Linear recurrences with varying coefficients, in Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grotschel and L. Lovasz, eds., Elsevier, Amsterdam, 1995, pp. 1135-1138.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

Cf. A001620, A047787, A097664, A097665-A097676.

SetDefaultRealField(RealField(100)); R:= RealField(); Exp(-Pi(R)/Sqrt(12))/Sqrt(3); // G. C. Greubel, Sep 07 2018

RealDigits[1/Sqrt[3]*E^(-Pi/Sqrt[12]), 10, 105][[1]] (* Robert G. Wilson v, Aug 28 2004 *)

PARI
```
3*exp(psi(1/3)+Euler)
```

Equals exp(-Pi/sqrt(12))/sqrt(3).

More terms from Robert G. Wilson v, Aug 28 2004

Offset corrected by R. J. Mathar, Feb 05 2009

A200134 Decimal expansion of the negated value of the digamma function at 3/4.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(3/4) = -1.085860879786472169626886762817...

G. C. Greubel, Table of n, a(n) for n = 1..10000
E. D. Krupnikov, K. S. Kölbig, Some special cases of the generalized hypergeometric function (q+1)Fq, J. Comp. Appl. Math. 78 (1997) 79-95.
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A001620, A020759, A020777, A047787, A301816.

SetDefaultRealField(RealField(100)); R:= RealField(); -EulerGamma(R) + Pi(R)/2 - 3*Log(2); // G. C. Greubel, Aug 29 2018

Maple
```
evalf(-gamma+Pi/2-3*log(2)) ;
```

RealDigits[ -PolyGamma[3/4], 10, 97] // First (* Jean-François Alcover, Feb 20 2013 *)
N[StieltjesGamma[0, 3/4], 99] (* Peter Luschny, May 16 2018 *)

-psi(3/4) \\ Charles R Greathouse IV, Nov 22 2011

Psi(3/4) = -gamma + Pi/2 - 3*log(2) = A000796 - A020777 = 3.14159... - 4.22745...

Pi = gamma(0,1/4) - gamma(0,3/4) = A020777 - A200134, where gamma(n,x) denotes the generalized Stieltjes constants. - Peter Luschny, May 16 2018

A200135 Decimal expansion of the negated value of the digamma function at 1/5.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			Psi(1/5) =  -5.289039896592188295547207962...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200064, A200134, A200136, A200137, A200138.

SetDefaultRealField(RealField(100)); R:= RealField(); -EulerGamma(R) -Pi(R)*Sqrt(1+2/Sqrt(5))/2 -5*Log(5)/4 -Sqrt(5)/4*Log((3+Sqrt(5)/2) ); // G. C. Greubel, Sep 03 2018

-gamma-Pi*sqrt(1+2/sqrt(5))/2-5*log(5)/4-sqrt(5)/4*log((3+sqrt(5)/2) ); evalf(%) ;

RealDigits[-PolyGamma[1/5], 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

-psi(1/5) \\ Charles R Greathouse IV, Jul 19 2013

Psi(1/5) = -gamma - Pi*sqrt(1 + 2/sqrt(5))/2 - 5*log(5)/4 -sqrt(5)*log((3 + sqrt(5))/2)/4 where gamma = A001620, sqrt(1 + 2/sqrt(5)) = A019952, (3 + sqrt(5))/2 = A104457.

More terms from Jean-François Alcover, Feb 11 2013

A200064 Decimal expansion of the negated value of the digamma function at 2/3.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Nov 13 2011

			psi(2/3) = -1.3182344157865884724023408166...

Cf. A020759, A047787, A020777

RealDigits[ -PolyGamma[2/3], 10, 98] // First (* Jean-François Alcover, Feb 20 2013 *)

-psi(2/3) \\ Charles R Greathouse IV, Nov 22 2011

psi(2/3) = -gamma+Pi*sqrt(3)/6-3*log(3)/2 = A093602 - A047787 = 1.813799 -3.132033...

A222457 Decimal expansion of the negated value of the digamma function at 1/6.

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Feb 21 2013

			Psi(1/6) = -6.3321275053749147924249615748457777225904948...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Ernst D. Krupnikov and K. S. Kolbig, Some special cases of the generalized hypergeometric function _{q+1}F_q, J. Comp. Appl. Math. 78 (1997) 79-95
Eric Weisstein's World of Mathematics, Gauss's Digamma Theorem.
Wikipedia, Digamma function: special values.
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A200064, A222458.

RealDigits[-PolyGamma[1/6], 10, 90][[1]]

Maxima
```
fpprec:90; ev(bfloat(-psi[0](1/6)));
```
PARI
```
-psi(1/6)
```

Psi(1/6) = -gamma -Pi*sqrt(3)/2 -3*log(3)/2 -2*log(2).

A200138 Decimal expansion of the negated value of the digamma function at 4/5.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Nov 13 2011

			Psi(4/5) = -0.965008566706138459391297633...

Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200134-A200137.

-gamma+Pi*sqrt(1+2/sqrt(5))/2-5*log(5)/4-sqrt(5)/4*log(3/2+sqrt(5)/2) ; evalf(%) ;

RealDigits[ -PolyGamma[4/5], 10, 87] // First (* Jean-François Alcover, Feb 20 2013 *)

-psi(4/5) \\ Charles R Greathouse IV, Nov 22 2011

Psi(4/5) = -gamma + Pi*sqrt(1+2/sqrt 5)/2 -5*log(5)*log((3+sqrt 5)/2)/4.

A222458 Decimal expansion of the negated value of the digamma function at 5/6.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Feb 21 2013

			Psi(5/6) = -0.890729412672261240642726801919310525738296...

Vincenzo Librandi, Table of n, a(n) for n = 0..5000
Eric Weisstein's World of Mathematics, Gauss's Digamma Theorem.
Wikipedia, Digamma function: Gauss's digamma theorem.
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A200064, A222457.

RealDigits[-PolyGamma[5/6], 10, 90][[1]]

Maxima
```
fpprec:90; ev(bfloat(-psi[0](5/6)));
```
PARI
```
-psi(5/6)
```

Psi(5/6) = -gamma + Pi*sqrt(3)/2 - 3*log(3)/2 - 2*log(2).

A250129 Decimal expansion of the negated value of the digamma function at 1/8.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jan 15 2015

			Psi(1/8) = -8.388492663295854867802742923086343000051446...

Eric Weisstein's MathWorld, Gauss's Digamma Theorem.
Wikipedia, Digamma function: special values.
Index entries for sequences related to the digamma function

Cf. A020759, A020777, A047787, A200064, A222457, A222458.

RealDigits[PolyGamma[1/8], 10, 105] // First

-psi(1/8) \\ Charles R Greathouse IV, Jan 15 2015

Psi(1/8) = -gamma - (1/2)*(1+sqrt(2))*Pi - sqrt(2)*arccoth(sqrt(2)) - 4*log(2).

A306716 Decimal expansion of the negated value of the digamma function at 1/10.

Original entry on oeis.org

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

Offset: 2

Vaclav Kotesovec, Aug 22 2019

			Equals 10.4237549404110767951682162190100254042916425624441889203263920841...

Eric Weisstein's World of Mathematics, Digamma Function
Wikipedia, Digamma function
Index entries for sequences related to the digamma function

Cf. A020759, A047787, A020777, A200135, A222457, A250129.

Maple
```
evalf(-Psi(1/10), 102);
```

RealDigits[-PolyGamma[1/10], 10, 105][[1]]

PARI
```
-psi(1/10)
```

Psi(1/10) = -gamma - Pi*5^(1/4)*(sqrt(2 + sqrt(5))/2) - 2*log(2) - 5*log(5)/4 - 3*sqrt(5)*log((1 + sqrt(5))/2)/2, where gamma is the Euler-Mascheroni constant A001620.

Equals gamma - H(-9/10), H(z) the harmonic number. - Peter Luschny, Aug 22 2019

cp's OEIS Frontend

A256425 Decimal expansion of the generalized Euler constant gamma(1,3).

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A097663 Decimal expansion of the constant 3*exp(psi(1/3) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A200134 Decimal expansion of the negated value of the digamma function at 3/4.

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A200135 Decimal expansion of the negated value of the digamma function at 1/5.

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A200064 Decimal expansion of the negated value of the digamma function at 2/3.

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A222457 Decimal expansion of the negated value of the digamma function at 1/6.

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A200138 Decimal expansion of the negated value of the digamma function at 4/5.