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%I #60 Feb 26 2022 04:25:48
%S 3,81,1215,15309,177147,1948617,20726199,215233605,2195382771,
%T 22082967873,219667417263,2165293113021,21182215236075,
%U 205891132094649,1990280943581607,19147875284802357,183448998696332259,1751104078464989745,16660504517966902431
%N a(n) = (2n-1) * 3^(2n-1).
%C Denominators of odd terms in expansion of arctanh(s/3); numerators are all 1. - _Gerry Martens_, Jul 26 2015
%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 28-40.
%H Harry J. Smith, <a href="/A060851/b060851.txt">Table of n, a(n) for n = 1..200</a>
%H Xavier Gourdon and Pascal Sebah, <a href="http://numbers.computation.free.fr/Constants/Miscellaneous/zeta.html">Riemann's zeta function</a>.
%H Pablo A. Panzone, <a href="http://www.scielo.org.ar/pdf/ruma/v50n1/v50n1a13.pdf">Formulas for the Euler-Mascheroni constant</a>, Rev. Un. Mat. Argentina, Vol 50, No. 1 (2009), pp. 161-164.
%H Simon Plouffe, <a href="http://www.numberworld.org/y-cruncher/records.html">Other interesting computations</a> at numberworld.org.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (18,-81).
%F Sum_{n>=1} 2/a(n) = log(2).
%F Sum_{n>=1} (2/a(n) - zeta(2n+1)/(2^(2n)*(2n+1))) = gamma (Euler's constant).
%F Sum_{n>=1} ((4n+2)/a(n) - zeta(2n+1)/2^(2n))/(2n+1) = gamma (Euler's constant).
%F Sum_{n>=1} ((4n+2)/a(n) - zeta(2n+1)/2^(2n)) = 7/4.
%F Sum_{n>=1} ((2n+1)/a(n) - zeta(2n+1)/2^(2n+1)) = 7/8.
%F From _R. J. Mathar_, May 07 2013: (Start)
%F G.f.: 3*x*(1+9*x) / (9*x-1)^2.
%F a(n+1) = 3*A155988(n). (End)
%F Sum_{n>=1} (-1)^(n+1)/a(n) = arctan(1/3). - _Amiram Eldar_, Feb 26 2022
%p A060851:=n->(2*n-1)*3^(2*n-1); seq(A060851(n), n=1..20); # _Wesley Ivan Hurt_, Dec 02 2013
%t Table[(2*n - 1)*3^(2*n - 1), {n, 20}] (* _Wesley Ivan Hurt_, Dec 02 2013 *)
%t a[n_] := 1/SeriesCoefficient[ArcTanh[s/3],{s,0,n}]
%t Table[a[n], {n, 1, 40, 2}] (* _Gerry Martens_, Jul 26 2015 *)
%o (PARI) for (n=1, 200, write("b060851.txt", n, " ", (2*n - 1)*(3^(2*n - 1))); ) \\ _Harry J. Smith_, Jul 13 2009
%o (Magma) [ (2*n-1) * (3^(2*n-1)): n in [1..100]]; // _Vincenzo Librandi_, Apr 20 2011
%Y Cf. A002162 (log(2)), A001620 (Euler's constant).
%K nonn,easy
%O 1,1
%A _Frank Ellermann_, May 03 2001
%E More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001
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