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A294830 Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(5*k+3) = A147874(k+2) for k >= 0.

%I A294830 #13 Feb 16 2025 08:33:51
%S A294830 4,8,1,7,0,1,7,7,4,4,9,5,8,2,8,7,7,7,0,7,7,0,7,5,9,2,9,3,6,1,9,1,4,7,
%T A294830 5,5,2,3,4,1,8,7,4,5,9,3,7,4,8,4,1,8,0,4,7,3,0,4,5,9,0,1,4,1,8,8,1,5,
%U A294830 0,5,5,7,2,3,1,7,1,8,8,9,7,5,6,8,1,9,7,7,0,2,2,1,4,0,1,6,0,3,5
%N A294830 Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(5*k+3) = A147874(k+2) for k >= 0.
%C A294830 In the Koecher reference v_5(3) = (2/5)*(present value V(5,3)) = 0.192680709798338..., given there as (1/4)*log(5) - (1/(2*sqrt(5)))*log((1 + sqrt(5))/2) - (Pi/10)*sqrt((5 - 2*sqrt(5))/5).
%D A294830 Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.
%H A294830 G. C. Greubel, <a href="/A294830/b294830.txt">Table of n, a(n) for n = 0..10000</a>
%H A294830 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>
%F A294830 Sum_{k>=0} 1/((5*n + 3)*(n + 1)) =: V(5,3) = ((5/2)*log(5) - (2*phi-1)*(log(phi) + (Pi/5)*sqrt(7-4*phi)))/4 = (1/2)*(-Psi(3/5) + Psi(1)) with the golden section phi =(1 + sqrt(5))/2 = A001622 with the digamma function Psi and Psi(1) = -gamma = A001620.
%F A294830 The partial sums of this series are given in A294828/A294829.
%e A294830 0.481701774495828777077075929361914755234187459374841804730459014188150...
%t A294830 RealDigits[((5/2)*Log[5] - (2*GoldenRatio - 1)*(Log[GoldenRatio] + (Pi/5)*Sqrt[7 - 4*GoldenRatio]))/4, 10, 100][[1]] (* _G. C. Greubel_, Aug 30 2018 *)
%o A294830 (PARI) default(realprecision, 100); phi=(1+sqrt(5))/2; ((5/2)*log(5) - (2*phi-1)*(log(phi) + (Pi/5)*sqrt(7-4*phi)))/4 \\ _G. C. Greubel_, Aug 30 2018
%o A294830 (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); phi:= (1 + Sqrt(5))/2; ((5/2)*Log(5) - (2*phi-1)*(Log(phi) + (Pi(R)/5)*Sqrt(7 - 4*phi)))/4; // _G. C. Greubel_, Aug 30 2018
%Y A294830 Cf. A001620, A001622, A147874, A294828/A294829.
%K A294830 nonn,cons
%O A294830 0,1
%A A294830 _Wolfdieter Lang_, Nov 16 2017