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Decimal expansion of the sum of the reciprocals of the 9-gonal (or enneagonal or nonagonal) numbers (A001106).
7

%I #29 Apr 22 2024 07:28:54

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

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

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

%N Decimal expansion of the sum of the reciprocals of the 9-gonal (or enneagonal or nonagonal) numbers (A001106).

%H Vincenzo Librandi, <a href="/A244646/b244646.txt">Table of n, a(n) for n = 1..1000</a>

%H Ravi P. Agarwal, <a href="https://doi.org/10.4236/jamp.2021.98132">Pythagoreans Figurative Numbers: The Beginning of Number Theory and Summation of Series</a>, Journal of Applied Mathematics and Physics, Vol.9, No.8 (2021), pp. 2038-2113. See p. 2076.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Polygonal_number">Polygonal number</a>.

%F Equals Sum_{n>=1} 2/(7n^2 - 5n).

%F Equals (2*log(14) + 4*(cos(Pi/7)*log(cos(3*Pi/14)) + log(sin(Pi/7))*sin(Pi/14) - log(cos(Pi/14)) * sin(3*Pi/14)) + Pi*tan(3*Pi/14))/5. - _Vaclav Kotesovec_, Jul 04 2014

%F Equals 14/25 - (2/5)*(gamma + psi(-5/7)), where gamma is Euler's constant (A001620) and psi(x) is the digamma function (Agarwal, 2021), psi(-5/7) = psi(2/7)+7/5 = -2.285517..., see A354628. - _Amiram Eldar_, Nov 12 2021

%e 1.2433209261537129892066077396310142821358441010300996244152817525...

%t RealDigits[ Sum[2/(7n^2 - 5n), {n, 1 , Infinity}], 10, 111][[1]]

%Y Cf. A000038, A001106, A013661, A001620, A244639, A244644, A244645, A244647, A244648, A244649.

%K nonn,cons,easy

%O 1,2

%A _Robert G. Wilson v_, Jul 03 2014