A294966 Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(6*k+5) = A049452(k+1) for k >= 0.

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, 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, 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

Offset: 0

Author

Wolfdieter Lang, Nov 27 2017

Keywords

• nonn
• cons

Comments

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)).

Examples

			0.313513747770728380036214711836908094696136733315523822488574116360843...
		

References

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

Links

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

Crossrefs

Cf. A001620, A049452, A294964/A294965.

Programs

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

Mathematica

RealDigits[-PolyGamma[0, 5/6] + PolyGamma[0, 1], 10, 100][[1]] (* G. C. Greubel, Sep 05 2018 *)

PARI

default(realprecision, 100); (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) \\ G. C. Greubel, Sep 05 2018

Formula

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.

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

Equals Sum_{k>=2} zeta(k)/6^(k-1). - Amiram Eldar, May 31 2021