%I #21 Jan 07 2024 01:54:01

%S 7,8,8,6,3,1,3,9,0,2,0,2,0,0,2,3,6,7,4,4,3,8,8,0,8,1,9,8,3,8,9,7,6,6,

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

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

%N Decimal expansion of the generalized Euler constant gamma(1,8).

%H G. C. Greubel, <a href="/A256781/b256781.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/8 + 1/8*(Pi/2*(sqrt(2)+1) + log(2) + sqrt(2)*log(sqrt(2) + 1)).

%F Equals Sum_{n>=0} (1/(8n+1) - 1/4*arctanh(4/(8n+5))).

%F Equals -(psi(1/8) + log(8))/8 = -(A250129 + A016631)/8. - _Amiram Eldar_, Jan 07 2024

%e 0.788631390202002367443880819838976661978118204921...

%t RealDigits[-3/8*Log[2] - PolyGamma[1/8]/8, 10, 105] // First

%o (PARI) Euler/8 + 1/8*(Pi/2*(sqrt(2)+1) + log(2) + sqrt(2)*log(sqrt(2) + 1)) \\ _Michel Marcus_, Apr 10 2015

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/8 + (1/8)*(Pi(R)/2*(Sqrt(2)+1) + Log(2) + Sqrt(2)*Log(Sqrt(2) + 1)); // _G. C. Greubel_, Aug 28 2018

%Y Cf. A001620 (EulerGamma), A016631, A228725 (gamma(1,2)), A250129, A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Apr 10 2015