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Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).
5

%I #20 Apr 10 2024 09:33:43

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

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

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

%N Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).

%H G. C. Greubel, <a href="/A244641/b244641.txt">Table of n, a(n) for n = 1..10000</a>

%H Hongwei Chen and G. C. Greubel, <a href="https://web.archive.org/web/20160305012605/http://www.siam.org/journals/categories/07-003.php">Sum of the Reciprocals of Polygonal Numbers (Solved)</a>, SIAM Problems and solutions.

%H Hongwei Chen and G. C. Greubel, <a href="/A244641/a244641.pdf">Siam, Problems and Solutions, problem 07-003 and the solution</a>

%F Sum_{n>=1} 2/(3*n^2 - n).

%F Equals 3*log(3) - Pi*sqrt(3)/3 = A016650 - A093602. - _Michel Marcus_, Jul 03 2014

%e 1.482037501770111223591657453125421381658405425375507779634198065524359698529473...

%t RealDigits[Sum[2/(3*n^2-n), {n,1,Infinity}], 10, 111][[1]]

%t RealDigits[3*Log[3] - Pi*Sqrt[3]/3, 10, 140][[1]] (* _G. C. Greubel_, Mar 24 2024 *)

%o (Magma) SetDefaultRealField(RealField(139)); R:= RealField(); 3*Log(3)-Pi(R)*Sqrt(3)/3; // _G. C. Greubel_, Mar 24 2024

%o (SageMath) numerical_approx(3*log(3)-pi*sqrt(3)/3, digits=139) # _G. C. Greubel_, Mar 24 2024

%Y Cf. A000038, A013661, A016627, A016650, A093602, A244639, A244645, A275792.

%K nonn,cons,easy

%O 1,2

%A _Robert G. Wilson v_, Jul 03 2014