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Decimal expansion of the generalized Euler constant gamma(1,5).
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%I #21 Jan 07 2024 01:54:32

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

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

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

%N Decimal expansion of the generalized Euler constant gamma(1,5).

%H G. C. Greubel, <a href="/A256779/b256779.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+1) - 2/5*arctanh(5/(10n+7))).

%F Equals -(psi(1/5) + log(5))/5 = (A200135 - A016628)/5. - _Amiram Eldar_, Jan 07 2024

%e 0.735920396831617584189289725844752893059997383987625...

%t RealDigits[-Log[5]/5 - PolyGamma[1/5]/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, A200135, A228725 (gamma(1,2)), A256425 (gamma(1,3)), A256778-A256784 (selection of ruler-and-compass constructible gamma(r,k)).

%K nonn,cons,easy

%O 0,1

%A _Jean-François Alcover_, Apr 10 2015