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%I #20 Jan 07 2024 01:54:19
%S 1,9,0,3,8,9,3,2,6,4,3,0,2,0,3,1,5,4,2,2,5,9,8,3,2,2,9,7,6,4,2,6,8,1,
%T 6,3,2,6,0,1,5,1,9,4,8,4,4,8,4,5,8,4,8,7,0,6,4,2,6,1,1,5,6,7,4,7,6,8,
%U 6,4,1,1,0,4,4,5,7,6,7,2,3,8,6,8,4,0,5,3,6,2,8,5,2,0,8,6,8,4,1,3,2,2,5,6,1
%N Decimal expansion of the generalized Euler constant gamma(2,5).
%H G. C. Greubel, <a href="/A256780/b256780.txt">Table of n, a(n) for n = 0..10000</a>
%H D. H. Lehmer, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27121.pdf">Euler constants for arithmetic progressions</a>, Acta Arith. 27 (1975), p. 134.
%F Equals EulerGamma/5 + Pi/10*sqrt(1 - 2/sqrt(5)) + log(5)/20 - sqrt(5)/10*log((1 + sqrt(5))/2).
%F Equals Sum_{n>=0} (1/(5n+2) - 2/5*arctanh(5/(10n+9))).
%F Equals -(psi(2/5) + log(5))/5 = (A200136 - A016628)/5. - _Amiram Eldar_, Jan 07 2024
%e 0.190389326430203154225983229764268163260151948448458487...
%t RealDigits[-Log[5]/5 - PolyGamma[2/5]/5, 10, 105] // First
%o (PARI) Euler/5 + Pi/10*sqrt(1 - 2/sqrt(5)) + log(5)/20 - sqrt(5)/10*log((1 + sqrt(5))/2) \\ _Michel Marcus_, Apr 10 2015
%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/5 + Pi(R)/10*Sqrt(1 - 2/Sqrt(5)) + Log(5)/20 - Sqrt(5)/10*Log((1 + Sqrt(5))/2); // _G. C. Greubel_, Aug 28 2018
%Y Cf. A001620 (EulerGamma), A016628, A200136, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).
%K nonn,cons,easy
%O 0,2
%A _Jean-François Alcover_, Apr 10 2015
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