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%I #7 Aug 04 2014 20:57:45
%S 17,44,83,134,197,272,359,458,569,692,827,974,1133,1304,1487,1682,
%T 1889,2108,2339,2582,2837,3104,3383,3674,3977,4292,4619,4958,5309,
%U 5672,6047,6434,6833,7244,7667,8102,8549,9008,9479,9962,10457,10964,11483,12014,12557
%N Floor(1/s(n)), where s(n) = (2n+1)/(2n+2) - n*log((n+1)/n).
%C 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 Clark Kimberling, <a href="/A227721/b227721.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = 2 + 9*n + 6*n^2 (conjectured).
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) (conjectured).
%F G.f.: (-17 + 7 x - 2 x^2)/(-1 + x)^3 (conjectured).
%t s[n_] := s[n] = (2 n + 1)/(2 n + 2) - n*Log[1 + 1/n]
%t Table[Floor[1/s[n]], {n, 1, 100}] (* A227721 *)
%t Table[Round[1/s[n]], {n, 1, 100}] (* conjecture: A094159 *)
%Y Cf. A227719, A227720, A094159.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Jul 22 2013
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