A256783 - OEIS

%I #16 Jan 07 2024 01:53:06

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

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

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

%N Decimal expansion of the generalized Euler constant gamma(1,12).

%H G. C. Greubel, <a href="/A256783/b256783.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/12 + 1/24*(Pi*(2+sqrt(3)) - 2*(sqrt(3)-1)*log(2) + log(3) + 4*sqrt(3) * log(sqrt(3)+1)).

%F Equals Sum_{n>=0} (1/(12n+1) - 1/12*log((12n+13)/(12n+1))).

%F Equals -(psi(1/12) + log(12))/12. - _Amiram Eldar_, Jan 07 2024

%e 0.83024988988662433938903419703214965055579639279727496201543...

%t RealDigits[-Log[12]/12 - PolyGamma[1/12]/12, 10, 103] // First

%o (PARI) default(realprecision, 100); Euler/12 + 1/24*(Pi*(2+sqrt(3)) - 2*(sqrt(3)-1)*log(2) + log(3) + 4*sqrt(3)*log(sqrt(3)+1)) \\ _G. C. Greubel_, Aug 28 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/12 + (1/24)*(Pi(R)*(2+Sqrt(3)) - 2*(Sqrt(3)-1)*Log(2) + Log(3) + 4*Sqrt(3)*Log(Sqrt(3)+1)); // _G. C. Greubel_, Aug 28 2018

%Y Cf. A001620 (EulerGamma), A016635, 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