OFFSET
1,1
COMMENTS
Denominators of odd terms in expansion of arctanh(s/3); numerators are all 1. - Gerry Martens, Jul 26 2015
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 28-40.
LINKS
Harry J. Smith, Table of n, a(n) for n = 1..200
Xavier Gourdon and Pascal Sebah, Riemann's zeta function.
Pablo A. Panzone, Formulas for the Euler-Mascheroni constant, Rev. Un. Mat. Argentina, Vol 50, No. 1 (2009), pp. 161-164.
Simon Plouffe, Other interesting computations at numberworld.org.
Index entries for linear recurrences with constant coefficients, signature (18,-81).
FORMULA
Sum_{n>=1} 2/a(n) = log(2).
Sum_{n>=1} (2/a(n) - zeta(2n+1)/(2^(2n)*(2n+1))) = gamma (Euler's constant).
Sum_{n>=1} ((4n+2)/a(n) - zeta(2n+1)/2^(2n))/(2n+1) = gamma (Euler's constant).
Sum_{n>=1} ((4n+2)/a(n) - zeta(2n+1)/2^(2n)) = 7/4.
Sum_{n>=1} ((2n+1)/a(n) - zeta(2n+1)/2^(2n+1)) = 7/8.
From R. J. Mathar, May 07 2013: (Start)
G.f.: 3*x*(1+9*x) / (9*x-1)^2.
a(n+1) = 3*A155988(n). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = arctan(1/3). - Amiram Eldar, Feb 26 2022
E.g.f.: (1 + exp(9*x)*(18*x - 1))/3. - Stefano Spezia, Dec 26 2024
MAPLE
MATHEMATICA
Table[(2*n - 1)*3^(2*n - 1), {n, 20}] (* Wesley Ivan Hurt, Dec 02 2013 *)
a[n_] := 1/SeriesCoefficient[ArcTanh[s/3], {s, 0, n}]
Table[a[n], {n, 1, 40, 2}] (* Gerry Martens, Jul 26 2015 *)
PROG
(PARI) a(n) = (2*n - 1)*(3^(2*n - 1)) \\ Harry J. Smith, Jul 13 2009
(Magma) [ (2*n-1) * (3^(2*n-1)): n in [1..100]]; // Vincenzo Librandi, Apr 20 2011
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Frank Ellermann, May 03 2001
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), May 07 2001
STATUS
approved