A294833 Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(5*k+4) = 2*A005476(k+1) for k >= 0.

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, 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, 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

Offset: 0

Author

Wolfdieter Lang, Nov 18 2017

Keywords

• nonn
• cons

Comments

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).

Examples

			0.387792901804605598784785543074432988592001153755299230304043559360093480608...
		

References

• Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193. For v_5(4) see p. 192.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Eric Weisstein's World of Mathematics, Digamma Function.

Crossrefs

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

Programs

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

Mathematica

RealDigits[-PolyGamma[0, 4/5] + PolyGamma[0, 1], 10, 100][[1]] (* G. C. Greubel, Sep 05 2018 *)

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

Formula

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.

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

Equals Sum_{k>=2} zeta(k)/5^(k-1). - Amiram Eldar, May 31 2021