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%I #19 Feb 08 2023 23:03:54
%S 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,
%T 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,
%U 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
%N Decimal expansion of the sum of the reciprocals of the Dodecagonal numbers (A051624).
%C From _Wolfdieter Lang_, Nov 09 2017: (Start)
%C 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.
%C The partial sums are given in A294520/A294521. (End)
%D Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.
%H Lawrence Downey, Boon W. Ong, and James 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. 5 (2008), 391-394.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Polygonal_number">Polygonal number</a>
%F Equals Sum_{n>=1} 1/(5n^2 - 4n).
%F Equals Pi/8*sqrt(1+2/sqrt(5)) + (5*log(5) + sqrt(5)*log((3+sqrt(5))/2))/16. - _Vaclav Kotesovec_, Jul 04 2014
%F 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
%F ((5/2)*log(5) + (2*phi - 1)*(log(phi) + (Pi/5)*sqrt(3 + 4*phi)))/8. - _Wolfdieter Lang_, Nov 09 2017
%e 1.1779560579226638587351739680918874184458572345666798028425228573...
%t RealDigits[ Sum[1/(5n^2 - 4n), {n, 1 , Infinity}], 10, 111][[1]]
%Y Cf. A000038, A013661, A051624, A244639, A244644, A244645, A244646, A244647, A244648, A294520/A294521.
%K nonn,cons,easy
%O 1,3
%A _Robert G. Wilson v_, Jul 03 2014
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