%I #18 Sep 08 2022 08:46:20

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

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

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

%N Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(6*k+5) = A049452(k+1) for k >= 0.

%C In the Koecher reference v_6(5) = (1/6)*(present value V(6,5)) = 0.05225229129512..., given on p. 192 as (1/4)*log(3) + (1/3)*log(2) - Pi/(4*sqrt(3)).

%D Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193.

%H G. C. Greubel, <a href="/A294966/b294966.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>.

%F Sum_{k>=0} 1/((6*n + 5)*(n + 1)) =: V(6,5) = (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) = -Psi(5/6) + Psi(1) with the digamma function Psi and Psi(1) = -gamma = A001620.

%F The partial sums of this series are given in A294964/A294965.

%F Equals Sum_{k>=2} zeta(k)/6^(k-1). - _Amiram Eldar_, May 31 2021

%e 0.313513747770728380036214711836908094696136733315523822488574116360843...

%t RealDigits[-PolyGamma[0, 5/6] + PolyGamma[0, 1], 10, 100][[1]] (* _G. C. Greubel_, Sep 05 2018 *)

%o (PARI) default(realprecision, 100); (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) \\ _G. C. Greubel_, Sep 05 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); (3/2)*Log(3) + 2*Log(2) - (1/2)*Pi(R)*Sqrt(3); // _G. C. Greubel_, Sep 05 2018

%Y Cf. A001620, A049452, A294964/A294965.

%K nonn,cons

%O 0,1

%A _Wolfdieter Lang_, Nov 27 2017