A097664 -id:A097664 - cp's OEIS frontend

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.

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

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.

Original entry on oeis.org

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

A097678 E.g.f.: (1/(1-x^3))exp( 3sum_{i>=0} x^(3i+2)/(3i+2) ) for an order-3 linear recurrence with varying coefficients.

Original entry on oeis.org

1, 0, 3, 6, 27, 252, 1125, 10206, 108297, 811944, 10272339, 131572350, 1410753267, 22363938324, 342373389813, 4790641828518, 90549635310225, 1626834238205904, 28073013793245603, 614304628556766966, 12727707975543382731

Offset: 0

Paul D. Hanna, Sep 01 2004

nonn

Limit_{n->inf} n*n!/a(n) = 3*c = 4.2896529252... where c = 3*exp(psi(2/3)+EulerGamma) = 1.4298843084...(A097664) and EulerGamma is the Euler-Mascheroni constant (A001620) and psi() is the Digamma function (see Mathworld link).

			The sequence {1, 0, 3/2!, 6/3!, 27/4!, 252/5!, 1125/6!, 10206/7!,...} is generated by a recursion described by Benoit Cloitre's generalized Euler-Gauss formula for the Gamma function (see Cloitre link).

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116. Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.
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.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
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.
Eric Weisstein's World of Mathematics, Digamma Function.

Cf. A097664, A097677, A097679-A097682.

CoefficientList[Series[1/Sqrt[(1-x^3)*(1-x)^3]*E^(-Sqrt[3] * ArcTan[Sqrt[3] * x/(2+x)]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 15 2014 *)

{a(n)=n!*polcoeff(1/(1-x^3)*exp(3*sum(i=0,n,x^(3*i+2)/(3*i+2)))+x*O(x^n),n)}

a(n)=if(n<0,0,if(n==0,1,3*(n-1)*a(n-2)+if(n<3,0,n!/(n-3)!*a(n-3))))

For n>=3: a(n) = 3*(n-1)*a(n-2) + n!/(n-3)!*a(n-3); a(0)=1, a(1)=0, a(2)=3. E.g.f.: 1/sqrt((1-x^3)*(1-x)^3)*exp(-sqrt(3)*atan(sqrt(3)*x/(2+x))).

cp's OEIS Frontend

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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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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Mathematica

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Extensions

A097678 E.g.f.: (1/(1-x^3))*exp( 3*sum_{i>=0} x^(3*i+2)/(3*i+2) ) for an order-3 linear recurrence with varying coefficients.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A097678 E.g.f.: (1/(1-x^3))exp( 3sum_{i>=0} x^(3i+2)/(3i+2) ) for an order-3 linear recurrence with varying coefficients.