A097666 -id:A097666 - cp's OEIS frontend

A097665 Decimal expansion of the constant 4*exp(psi(1/4) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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

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-4 linear recursions with varying coefficients (see A097679 for example).

			c = 0.10393978817538095427347780991748938501693892081588480403756...

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..2500
Benoit Cloitre, On a generalization of Euler-Gauss formula for the Gamma function, preprint 2004.
Andrew Odlyzko, Asymptotic enumeration methods, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.
Xavier Gourdon and Pascal Sebah, Introduction to the Gamma Function.

Cf. A097663-A097664, A097666-A097676.

RealDigits[1/2*E^(-Pi/2), 10, 105][[1]] (* Robert G. Wilson v, Aug 28 2004 *)

PARI
```
4*exp(psi(1/4)+Euler)
```

c = 1/2*exp(-Pi/2).

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

A097667 Decimal expansion of the constant 5*exp(psi(1/5) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

Original entry on oeis.org

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

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-5 linear recursions with varying coefficients (see A097680 for example).

			c = 0.04494182877920882060846739642766520340238594371059869805861...

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..2500
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. A097663-A097666, A097668-A097676.

RealDigits[ GoldenRatio^(-Sqrt[5]/2)/5^(1/4)*E^(-Pi/2*Sqrt[1 + 2/Sqrt[5]]), 10, 104][[1]] (* Robert G. Wilson v, Aug 27 2004 *)
Join[{0}, RealDigits[N[5*Exp[PolyGamma[1/5] + EulerGamma], 120], 10, 100][[1]]] (* G. C. Greubel, Dec 31 2016 *)

PARI
```
5*exp(psi(1/5)+Euler)
```

c = ((sqrt(5)+1)/2)^(-sqrt(5)/2)/5^(1/4)*exp(-Pi/2*sqrt(1+2/sqrt(5))).

More terms from Robert G. Wilson v, Aug 27 2004

A247718 Decimal expansion of Integral_{0..Pi/2} exp(t)*cos(t) dt.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Sep 23 2014

			1.90523869048267582773651783335191656319508543733226747...

G. C. Greubel, Table of n, a(n) for n = 1..10000
D. H. Bailey, J. M. Borwein, Highly Parallel, High-Precision Numerical Integration p. 6. (2005) Lawrence Berkeley National Laboratory.

Cf. A042972.

SetDefaultRealField(RealField(100)); R:= RealField(); (Exp(Pi(R)/2) - 1)/2; // G. C. Greubel, Sep 07 2018

RealDigits[(Exp[Pi/2] - 1)/2, 10, 105] // First

default(realprecision, 100); (exp(Pi/2) -1)/2 \\ G. C. Greubel, Sep 07 2018

Equals (A042972 - 1)/2 = A097666 -1/2, where A042972 is exp(Pi/2).

cp's OEIS Frontend

A097665 Decimal expansion of the constant 4*exp(psi(1/4) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A097667 Decimal expansion of the constant 5*exp(psi(1/5) + EulerGamma), where EulerGamma is the Euler-Mascheroni constant (A001620) and psi(x) is the digamma function.

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A247718 Decimal expansion of Integral_{0..Pi/2} exp(t)*cos(t) dt.

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Magma

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