A244649 Decimal expansion of the sum of the reciprocals of the Dodecagonal numbers (A051624).

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

Offset

1,3

Comments

From Wolfdieter Lang, Nov 09 2017: (Start)

In the Downey et al. link this is the instance k = 5 of the formula given there for S_{2*k+2}. A simpler formula is given in the Koecher reference as (5/4)*v_5(1) on p. 192. See the Kotesovec formula given below.

The partial sums are given in A294520/A294521. (End)

References

Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.

Links

Table of n, a(n) for n=1..105.

Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.

Wikipedia, Polygonal number

Formula

Equals Sum_{n>=1} 1/(5n^2 - 4n).

Equals Pi/8*sqrt(1+2/sqrt(5)) + (5*log(5) + sqrt(5)*log((3+sqrt(5))/2))/16. - Vaclav Kotesovec, Jul 04 2014

This is the value given in the Koecher reference (see a comment above), and rewritten with the golden section phi = (1 + sqrt(5))/2 this becomes

((5/2)*log(5) + (2*phi - 1)*(log(phi) + (Pi/5)*sqrt(3 + 4*phi)))/8. - Wolfdieter Lang, Nov 09 2017

Example

1.1779560579226638587351739680918874184458572345666798028425228573...

Mathematica

RealDigits[ Sum[1/(5n^2 - 4n), {n, 1 , Infinity}], 10, 111][[1]]

Crossrefs

Cf. A000038, A013661, A051624, A244639, A244644, A244645, A244646, A244647, A244648, A294520/A294521.

Sequence in context: A361010 A353231 A335847 * A267040 A225961 A099290

Adjacent sequences: A244646 A244647 A244648 * A244650 A244651 A244652

Keyword

nonn,cons,easy

Author

Robert G. Wilson v, Jul 03 2014

Status

approved