cp's OEIS Frontend

A205325 Decimal expansion of the limit of [0;1,1,...] + [0;2,2,...] + ... + [0;n,n,...] - log(n) as n approaches infinity.

%I A205325 #21 Jan 15 2014 10:48:19
%S A205325 0,4,1,6,6,6,2,6,2,7,6,3,4,8,4,8,1,0,8,7,0,1,1,6,3,5,8,5,6,9,2,3,2,0,
%T A205325 7,4,3,1,2,5,4,5,4,6,7,5,2,8,4,1,6,3,1,8,0,9,2,0,1,3,5,9,2,3,2,9,9,1,
%U A205325 6,4,5,7,7,5,1,2,6,2,5,5,3,7,8,3,9,5,0,3
%N A205325 Decimal expansion of the limit of [0;1,1,...] + [0;2,2,...] + ... + [0;n,n,...] - log(n) as n approaches infinity.
%H A205325 Martin Janecke, <a href="http://prlbr.de/2010/01/edle-reihe/">Edle Reihe</a>
%F A205325 lim_{n->infinity} (1/[1;1,...] + 1/[2;2,...] + 1/[3;3,...] + ... + 1/[n;n,...] - log(n)).
%F A205325 lim_{n->infinity} (sum_{k=1...n} (2/(k + sqrt(k^2 + 4))) - log(n)).
%e A205325 0.0416662....
%t A205325 digits = 10; dn = 1000000; Clear[f]; f[n_] := NSum[2/(k + Sqrt[k^2+4]) - 1/k, {k, 1, Infinity}, NSumTerms -> 200000, WorkingPrecision -> digits+10, Method -> {"EulerMaclaurin", Method -> {"NIntegrate", "MaxRecursion" -> 20}}] + EulerGamma // RealDigits[#, 10, digits+2]& // First; f[dn]; f[n = 2*dn]; While[f[n] != f[n-dn], n = n+dn]; Prepend[ f[n][[1 ;; digits]], 0] (* _Jean-François Alcover_, Feb 25 2013 *)
%Y A205325 Cf. A001620, A205326, continued fractions A001622, A014176, A098316, A098317, A098318.
%K A205325 cons,nonn
%O A205325 0,2
%A A205325 _Martin Janecke_, Jan 26 2012
%E A205325 More terms from _Jean-François Alcover_, Feb 25 2013
%E A205325 More terms from _Jon E. Schoenfield_, Jan 05 2014