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

1,2

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

Formula

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

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

Example

1.482037501770111223591657453125421381658405425375507779634198065524359698529473...

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 *)

Prog

(Magma) SetDefaultRealField(RealField(139)); R:= RealField(); 3*Log(3)-Pi(R)*Sqrt(3)/3; // G. C. Greubel, Mar 24 2024
(SageMath) numerical_approx(3*log(3)-pi*sqrt(3)/3, digits=139) # G. C. Greubel, Mar 24 2024

Crossrefs

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

Sequence in context: A295086 A331331 A134484 * A274192 A021958 A200412

Adjacent sequences: A244638 A244639 A244640 * A244642 A244643 A244644

Keyword

nonn,cons,easy

Author

Robert G. Wilson v, Jul 03 2014

Status

approved