A097667 - OEIS

%I #22 Feb 27 2021 13:20:38

%S 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,

%T 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,

%U 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

%N 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.

%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-5 linear recursions with varying coefficients (see A097680 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="/A097667/b097667.txt">Table of n, a(n) for n = 0..2500</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 c = ((sqrt(5)+1)/2)^(-sqrt(5)/2)/5^(1/4)*exp(-Pi/2*sqrt(1+2/sqrt(5))).

%e c = 0.04494182877920882060846739642766520340238594371059869805861...

%t 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 *)

%t Join[{0}, RealDigits[N[5*Exp[PolyGamma[1/5] + EulerGamma], 120], 10, 100][[1]]] (* _G. C. Greubel_, Dec 31 2016 *)

%o (PARI) 5*exp(psi(1/5)+Euler)

%Y Cf. A097663-A097666, A097668-A097676.

%K cons,nonn

%O 0,2

%A _Paul D. Hanna_, Aug 25 2004

%E More terms from _Robert G. Wilson v_, Aug 27 2004