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

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, 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, 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

Offset: 1

Author

Robert G. Wilson v, Jul 03 2014

Keywords

• nonn
• cons
• easy

Examples

			1.482037501770111223591657453125421381658405425375507779634198065524359698529473...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..10000
• Hongwei Chen and G. C. Greubel, Sum of the Reciprocals of Polygonal Numbers (Solved), SIAM Problems and solutions.
• Hongwei Chen and G. C. Greubel, Siam, Problems and Solutions, problem 07-003 and the solution

Crossrefs

Cf. A000326, A016650, A093602, A294514.

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).

Programs

Magma

SetDefaultRealField(RealField(139)); R:= RealField(); 3*Log(3)-Pi(R)*Sqrt(3)/3; // G. C. Greubel, Mar 24 2024

Mathematica

RealDigits[Sum[2/(3*n^2-n), {n,1,Infinity}], 10, 111][[1]]
RealDigits[3*Log[3] - Pi*Sqrt[3]/3, 10, 140][[1]] (* G. C. Greubel, Mar 24 2024 *)

SageMath

numerical_approx(3*log(3)-pi*sqrt(3)/3, digits=139) # G. C. Greubel, Mar 24 2024

Formula

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

Equals 3*log(3) - Pi*sqrt(3)/3 = A016650 - A093602. - Michel Marcus, Jul 03 2014

Equals 2*A294514. - Hugo Pfoertner, Apr 24 2025