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A247914 Least number k such that |(k+1)/u(k+1) - e| < 1/n^n, where u is defined as in Comments.

%I A247914 #4 Sep 27 2014 19:03:08
%S A247914 1,3,4,5,6,7,9,10,11,12,14,15,16,17,19,20,21,22,23,25,26,27,28,30,31,
%T A247914 32,33,34,36,37,38,39,40,42,43,44,45,46,48,49,50,51,52,54,55,56,57,58,
%U A247914 59,61,62,63,64,65,67,68,69,70,71,73,74,75,76,77,78,80
%N A247914 Least number k such that |(k+1)/u(k+1) - e| < 1/n^n, where u is defined as in Comments.
%C A247914 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 {1,2} for n >=  1, that d(n+1) - d(n) is in {-1,0,1}, and that similar bounds hold for higher differences.
%D A247914 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 19.
%H A247914 Clark Kimberling, <a href="/A247914/b247914.txt">Table of n, a(n) for n = 1..1000</a>
%e A247914 Approximations for the first few terms |(n+1)/u(n+1) - e| and 1/n^n are shown here:
%e A247914 n ... |(n+1)/u(n+1)-e| .. 1/n^n
%e A247914 1 ... 0.7182818285 ...... 1
%e A247914 2 ... 0.28171817 ........ 0.25
%e A247914 3 ... 0.051615161 ....... 0.037037
%e A247914 4 ... 0.0089908988 ...... 0.00390625
%e A247914 5 ... 0.0013006963 ...... 0.00032000
%e A247914 a(2) = 3 because |4/u(4) - e| < 1/2^2 < |3/u(3) - e|.
%t A247914 $RecursionLimit = Infinity; $MaxExtraPrecision = Infinity;
%t A247914 z = 500; u[1] = 0; u[2] = 1; u[n_] := u[n] = u[n - 1] + u[n - 2]/(n - 2);
%t A247914 f[n_] := f[n] = Select[Range[z], Abs[(# + 1)/u[# + 1] - E] < n^-n &, 1];
%t A247914 u = Flatten[Table[f[n], {n, 1, z}]]  (* A247914 *)
%t A247914 w = Differences[u]
%t A247914 f1 = Flatten[Position[w, 1]] (* A247915 *)
%t A247914 f2 = Flatten[Position[w, 2]] (* A247916 *)
%Y A247914 Cf. A247908, A247911, A247915, A247916.
%K A247914 nonn,easy
%O A247914 1,2
%A A247914 _Clark Kimberling_, Sep 27 2014