A244639 Decimal expansion of the sum of the reciprocals of the heptagonal numbers (A000566).

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

Offset

1,2

Comments

For the partial sums of one half of this series, that is Sum_{k>=0} 1/((k+1)*(5*k+2)), with value 0.6613896265611944283..., see A294826(n)/A294827(n), for n >= 0. - Wolfdieter Lang, Nov 16 2017

References

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

Links

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

L. Downey, B. W. Ong, and J. A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J. 39, no. 8, 2008, 391-394.

Society for Industrial and Applied Mathematics, Sums of Reciprocals of Polygonal Numbers and a Theorem of Gauss

Wikipedia, Heptagonal Number

Formula

Equals Sum_{n>=1} 2/(5n^2 - 3n).

((5/2)*log(5) - (2*phi-1)*(log(phi) - (Pi/5)*sqrt(7-4*phi)))/3, with the golden section phi := (1 + sqrt(5))/2. This is (5/10)*v_5(2) given from the Koecher reference on p. 192 as ((5/2)*log(5) - sqrt(5)*log((1+sqrt(5))/2) + (1/5)*Pi*sqrt(5*(5-2*sqrt(5))))/3. Compare this with the number given in the Mathematica program. - Wolfdieter Lang, Nov 16 2017

Example

1.32277925312238885674944226131008401652280117371392437228545762688516221076....

Mathematica

RealDigits[ Pi*Sqrt[25 - 10 Sqrt[5]]/15 + 2Log[5]/3 + (1 + Sqrt[5]) Log[ Sqrt[ 10 - 2 Sqrt[5]]/2]/3 + (1 - Sqrt[5]) Log[ Sqrt[ 10 + 2 Sqrt[5]]/2]/3, 10, 111][[1]] (* or *)
RealDigits[ Sum[2/(5 n^2 - 3 n), {n, 1, Infinity}], 10, 111][[1]]

Prog

(PARI) sumnumrat(2/n/(5*n-3), 1) \\ Charles R Greathouse IV, Feb 08 2023

Crossrefs

Sequence in context: A092419 A293268 A020835 * A352673 A193919 A055674

Adjacent sequences: A244636 A244637 A244638 * A244640 A244641 A244642

Keyword

nonn,cons,easy

Author

Robert G. Wilson v, Jul 03 2014

Status

approved