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A294966
Decimal expansion of the sum of the reciprocals of the numbers (k+1)*(6*k+5) = A049452(k+1) for k >= 0.
2
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
0,1
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)).
REFERENCES
Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193.
LINKS
Eric Weisstein's World of Mathematics, Digamma Function.
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
EXAMPLE
0.313513747770728380036214711836908094696136733315523822488574116360843...
MATHEMATICA
RealDigits[-PolyGamma[0, 5/6] + PolyGamma[0, 1], 10, 100][[1]] (* G. C. Greubel, Sep 05 2018 *)
PROG
(PARI) default(realprecision, 100); (3/2)*log(3) + 2*log(2) - (1/2)*Pi*sqrt(3) \\ G. C. Greubel, Sep 05 2018
(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
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Wolfdieter Lang, Nov 27 2017
STATUS
approved