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

Author

Robert G. Wilson v, Jul 03 2014

Keywords

• nonn
• cons
• easy

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)

Examples

			1.1779560579226638587351739680918874184458572345666798028425228573...
		

References

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

Links

• 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

Crossrefs

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

Programs

Mathematica

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

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