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A227720 Round(1/s(n)), where s(n) = n*log(1+1/n) - (2n-1)/(2n).

%I A227720 #5 Jul 24 2013 12:50:35
%S A227720 5,16,34,57,86,121,163,210,263,322,388,459,536,619,709,804,905,1012,
%T A227720 1126,1245,1370,1501,1639,1782,1931,2086,2248,2415,2588,2767,2953,
%U A227720 3144,3341,3544,3754,3969,4190,4417,4651,4890,5135,5386,5644,5907,6176,6451,6733
%N A227720 Round(1/s(n)), where s(n) = n*log(1+1/n) - (2n-1)/(2n).
%C A227720 That s(n) > 0 for n >=1 follows from the chain 1 < log 2 < 3/4 < 2 log 3/2 < 5/6 < 3 log 4/3 < 7/8 < 4 log 5/4 < ... ; i.e., n log((n+1)/n) - (2n-1)/(2n) > 0 and (2n+1)/(2n+2) - n log((n+1)/n) > 0.  For the first, closeness to 0 is indicated by A227719 and A227720, and for the second, by A227721 and a sequence which possibly equals A094159.  Conjecture:  the four sequences are linearly recurrent.
%H A227720 Clark Kimberling, <a href="/A227720/b227720.txt">Table of n, a(n) for n = 1..1000</a>
%F A227720 a(n) = -2*a(n-1) - a(n-2) + a(n-4) - 2*a(n-5) + a(n-6) (conjectured).
%F A227720 G.f.:  (-5 - 6 x - 7 x^2 - 5 x^3 - x^4)/((-1 + x)^3 (1 + x + x^2 + x^3))  (conjectured).
%t A227720 s[n_] := n*Log[1 + 1/n] - (2 n - 1)/(2 n);
%t A227720 Table[Floor[1/s[n]], {n, 1, 100}]  (* A227719 *)
%t A227720 Table[Round[1/s[n]], {n, 1, 100}]  (* A227720 *)
%Y A227720 Cf. A227719, A227721, A094159.
%K A227720 nonn,easy
%O A227720 1,1
%A A227720 _Clark Kimberling_, Jul 22 2013