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A244641 Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).

%I A244641 #27 Apr 24 2025 13:20:42
%S A244641 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 A244641 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 A244641 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 A244641 Decimal expansion of the sum of the reciprocals of the pentagonal numbers (A000326).
%H A244641 G. C. Greubel, <a href="/A244641/b244641.txt">Table of n, a(n) for n = 1..10000</a>
%H A244641 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 A244641 Hongwei Chen and G. C. Greubel, <a href="/A244641/a244641.pdf">Siam, Problems and Solutions, problem 07-003 and the solution</a>
%F A244641 Sum_{n>=1} 2/(3*n^2 - n).
%F A244641 Equals 3*log(3) - Pi*sqrt(3)/3 = A016650 - A093602. - _Michel Marcus_, Jul 03 2014
%F A244641 Equals 2*A294514. - _Hugo Pfoertner_, Apr 24 2025
%e A244641 1.482037501770111223591657453125421381658405425375507779634198065524359698529473...
%t A244641 RealDigits[Sum[2/(3*n^2-n), {n,1,Infinity}], 10, 111][[1]]
%t A244641 RealDigits[3*Log[3] - Pi*Sqrt[3]/3, 10, 140][[1]] (* _G. C. Greubel_, Mar 24 2024 *)
%o A244641 (Magma) SetDefaultRealField(RealField(139)); R:= RealField(); 3*Log(3)-Pi(R)*Sqrt(3)/3; // _G. C. Greubel_, Mar 24 2024
%o A244641 (SageMath) numerical_approx(3*log(3)-pi*sqrt(3)/3, digits=139) # _G. C. Greubel_, Mar 24 2024
%Y A244641 Cf. A000326, A016650, A093602, A294514.
%Y A244641 Decimal expansion of the sum of the reciprocals of the m-gonal numbers: A000038 (m=3), A013661 (m=4), this sequence (m=5), A016627 (m=6), A244639 (m=7), A244645 (m=8), A244646 (m=9), A244647 (m=10), A244648 (m=11), A244649 (m=12), A275792 (m=14).
%K A244641 nonn,cons,easy
%O A244641 1,2
%A A244641 _Robert G. Wilson v_, Jul 03 2014