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

%I A244639 #38 Feb 08 2023 15:16:44
%S A244639 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,
%T A244639 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,
%U A244639 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
%N A244639 Decimal expansion of the sum of the reciprocals of the heptagonal numbers (A000566).
%C A244639 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
%D A244639 Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.
%H A244639 L. Downey, B. W. Ong, and J. A. Sellers, <a href="https://www.d.umn.edu/~jsellers/downey_ong_sellers_cmj_preprint.pdf">Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers</a>, Coll. Math. J. 39, no. 8, 2008, 391-394.
%H A244639 Society for Industrial and Applied Mathematics, <a href="https://archive.siam.org/journals/problems/downloadfiles/07-003s.pdf">Sums of Reciprocals of Polygonal Numbers and a Theorem of Gauss</a>
%H A244639 Wikipedia, <a href="http://en.wikipedia.org/wiki/Heptagonal_number">Heptagonal Number</a>
%F A244639 Equals Sum_{n>=1} 2/(5n^2 - 3n).
%F A244639 ((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
%e A244639 1.32277925312238885674944226131008401652280117371392437228545762688516221076....
%t A244639 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 *)
%t A244639 RealDigits[ Sum[2/(5 n^2 - 3 n), {n, 1, Infinity}], 10, 111][[1]]
%o A244639 (PARI) sumnumrat(2/n/(5*n-3),1) \\ _Charles R Greathouse IV_, Feb 08 2023
%K A244639 nonn,cons,easy
%O A244639 1,2
%A A244639 _Robert G. Wilson v_, Jul 03 2014