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

Author

Robert G. Wilson v, Jul 03 2014

Keywords

• nonn
• cons
• easy

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

Examples

			1.32277925312238885674944226131008401652280117371392437228545762688516221076....
		

References

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

Links

• 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

Programs

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]]

PARI

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

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