A294833 - OEIS

%I #19 Sep 08 2022 08:46:20

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

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

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

%N Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(5*k+4) = 2*A005476(k+1) for k >= 0.

%C In the Koecher reference v_5(4) = (1/5)*(present value V(5,4)) = 0.07755858036092111..., 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 Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193. For v_5(4) see p. 192.

%H G. C. Greubel, <a href="/A294833/b294833.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/((5*n + 4)*(n + 1)) =: V(5,4) = ((5/2)*log(5) + (2*phi - 1)*(log(phi) - (Pi/5)*sqrt(3 + 4*phi)))/2 = -Psi(4/5) + Psi(1) with the golden section phi =(1 + sqrt(5))/2 = A001622 with the digamma function Psi and Psi(1) = -gamma = A001620.

%F The partial sums of this series are given in A294831/A294832.

%F Equals Sum_{k>=2} zeta(k)/5^(k-1). - _Amiram Eldar_, May 31 2021

%e 0.387792901804605598784785543074432988592001153755299230304043559360093480608...

%t RealDigits[-PolyGamma[0, 4/5] + PolyGamma[0, 1], 10, 100][[1]] (* _G. C. Greubel_, Sep 05 2018 *)

%o (PARI) default(realprecision, 100); phi=(1+sqrt(5))/2; ((5/2)*log(5) + (2*phi - 1)*(log(phi) - (Pi/5)*sqrt(3 + 4*phi)))/2 \\ _G. C. Greubel_, Sep 05 2018

%o (Magma) SetDefaultRealField(RealField(100)); R:= RealField(); ((5/2)*Log(5) + Sqrt(5)*(Log((1+Sqrt(5))/2) - (Pi(R)/5)*Sqrt(5+2*Sqrt(5))))/2; // _G. C. Greubel_, Sep 05 2018

%Y Cf. A001620, A001622, 2*A005476, A294831/A294832.

%K nonn,cons

%O 0,1

%A _Wolfdieter Lang_, Nov 18 2017