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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. 17

%I #29 Mar 08 2023 07:49:34

%S 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,

%T 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,

%U 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

%N 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.

%C 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).

%D 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.

%H G. C. Greubel, <a href="/A097663/b097663.txt">Table of n, a(n) for n = 0..5000</a>

%H Benoit Cloitre, <a href="/A097679/a097679.pdf">On a generalization of Euler-Gauss formula for the Gamma function</a>, preprint 2004.

%H Xavier Gourdon and Pascal Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/gammaFunction.html">Introduction to the Gamma Function</a>.

%H Andrew Odlyzko, <a href="http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.pdf">Asymptotic enumeration methods</a>, in Handbook of Combinatorics, vol. 2, 1995, pp. 1063-1229.

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

%e 0.23311909318456411730537562326544289574460858702592456414096...

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

%o (PARI) 3*exp(psi(1/3)+Euler)

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); Exp(-Pi(R)/Sqrt(12))/Sqrt(3); // _G. C. Greubel_, Sep 07 2018

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

%K cons,nonn

%O 0,1

%A _Paul D. Hanna_, Aug 25 2004

%E More terms from _Robert G. Wilson v_, Aug 28 2004

%E Offset corrected by _R. J. Mathar_, Feb 05 2009

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