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A244649 Decimal expansion of the sum of the reciprocals of the Dodecagonal numbers (A051624). 6
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 (list; constant; graph; refs; listen; history; text; internal format)
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, A244639, A244645, A244646, A244647, A244648, A294520/A294521.

Sequence in context: A157290 A021566 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

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Last modified November 29 17:58 EST 2021. Contains 349416 sequences. (Running on oeis4.)