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A256849 Decimal expansion of the generalized Euler constant gamma(4,5) (negated). 9

%I #13 Sep 08 2022 08:46:12

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

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

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

%N Decimal expansion of the generalized Euler constant gamma(4,5) (negated).

%H G. C. Greubel, <a href="/A256849/b256849.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 -log(5)/5 - PolyGamma[4/5)/5.

%F Equals EulerGamma/5 - Pi/(10*sqrt(2*(5-sqrt(5)))) - Pi/(2*sqrt(10*(5-sqrt(5)))) + log(5)/20 - log(5-sqrt(5))/(4*sqrt(5)) + log(5+sqrt(5))/( 4*sqrt(5)).

%e -0.12888586914559238304189234001387044397828817291465897856 ...

%t RealDigits[-Log[5]/5 - PolyGamma[4/5]/5, 10, 107] // First

%o (PARI) default(realprecision, 100); Euler/5 - Pi/(10*sqrt(2*(5-sqrt(5)))) - Pi/(2*sqrt(10*(5-sqrt(5)))) + log(5)/20 - log(5-sqrt(5))/(4*sqrt(5)) + log(5+sqrt(5))/( 4*sqrt(5)) \\ _G. C. Greubel_, Aug 28 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); EulerGamma(R)/5 - Pi(R)/(10*Sqrt(2*(5-Sqrt(5)))) - Pi(R)/(2*Sqrt(10*(5-Sqrt(5)))) + Log(5)/20 - Log(5-Sqrt(5))/(4*Sqrt(5)) + Log(5+Sqrt(5))/( 4*Sqrt(5)); // _G. C. Greubel_, Aug 28 2018

%Y Cf. A001620 (gamma(1,1) = EulerGamma),

%Y Primitive ruler-and-compass constructible gamma(r,k): A228725 (1,2), A256425 (1,3), A256778 (1,4), A256779 (1,5), A256780 (2,5), A256781 (1,8), A256782 (3,8), A256783 (1,12), A256784 (5,12),

%Y Other gamma(r,k) (1 <= r <= k <= 5): A239097 (2,2), A256843 (2,3), A256844 (3,3), A256845 (2,4), A256846 (3,4), A256847 (4,4), A256848 (3,5), A256849 (4,5), A256850 (5,5).

%K nonn,cons,easy

%O 0,2

%A _Jean-François Alcover_, Apr 11 2015

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)