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A247908 Least number k such that e - 2*k/u(2*k) < 1/n^n, where u is defined as in Comments.

%I A247908 #6 Sep 27 2014 19:02:25
%S A247908 1,2,3,3,4,4,5,6,6,7,8,8,9,9,10,11,11,12,12,13,14,14,15,16,16,17,17,
%T A247908 18,19,19,20,20,21,22,22,23,23,24,25,25,26,26,27,28,28,29,29,30,30,31,
%U A247908 32,32,33,33,34,35,35,36,36,37,38,38,39,39,40,41,41,42
%N A247908 Least number k such that e - 2*k/u(2*k) < 1/n^n, where u is defined as in Comments.
%C A247908 The sequence u is define recursively by u(n) = u(n-1) + u(n-2)/(n-2), with u(1) = 0 and u(2) = 1.  Let d(n) = a(n+1) - a(n).  It appears that d(n) is in {0,1} for n >=  1, that d(n+1) - d(n) is in {2,3}, and that similar bounds hold for higher differences.
%D A247908 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 19.
%H A247908 Clark Kimberling, <a href="/A247908/b247908.txt">Table of n, a(n) for n = 1..1000</a>
%e A247908 Approximations for the first few terms of e - 2*n/u(2*n) and 1/n^n are shown here:
%e A247908 n ... e-2*n/u(2*n) .... 1/n^n
%e A247908 1 ... 0.71828 ........  1
%e A247908 2 ... 0.0516152 ....... 0.25
%e A247908 3 ... 0.0013007 ....... 0.037037
%e A247908 4 ... 0.0000184967 .... 0.00390625
%e A247908 a(2) = 2 because e - 4/u(4) < 1/2^2 < e - 2/u(2).
%t A247908 $RecursionLimit = 1000; $MaxExtraPrecision = 1000;
%t A247908 z = 300; u[1] = 0; u[2] = 1; u[n_] := u[n] = u[n - 1] + u[n - 2]/(n - 2);
%t A247908 f[n_] := f[n] = Select[Range[z], E - 2 #/u[2 #] < 1/n^n &, 1];
%t A247908 u = Flatten[Table[f[n], {n, 1, z}]]  (* A247908 *)
%t A247908 w = Differences[u]
%t A247908 Flatten[Position[w, 0]] (* A247909 *)
%t A247908 Flatten[Position[w, 1]] (* A247910 *)
%Y A247908 Cf. A247909, A247910, A247911, A247914.
%K A247908 nonn,easy
%O A247908 1,2
%A A247908 _Clark Kimberling_, Sep 27 2014