cp's OEIS Frontend

A069887 Number of terms in the simple continued fraction expansion for (1+1/n)^n.

%I A069887 #27 May 06 2022 07:35:15
%S A069887 1,2,5,7,7,10,14,16,24,16,20,29,39,40,42,39,46,42,44,57,59,55,66,55,
%T A069887 57,70,68,81,86,81,91,109,106,108,119,117,123,118,124,118,120,133,142,
%U A069887 147,164,155,159,164,167,163,177,176,168,171,198,198,201,201,205,206,227
%N A069887 Number of terms in the simple continued fraction expansion for (1+1/n)^n.
%C A069887 Limit_{n -> infinity} (1+1/n)^n = e.
%C A069887 For any natural number N, limit_{n->infinity} (log(N)^(1/n) + 1/n)^n = e*log(N). - _Alexander R. Povolotsky_, Dec 06 2007
%H A069887 Amiram Eldar, <a href="/A069887/b069887.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from G. C. Greubel)
%H A069887 Amiram Eldar, <a href="/A069887/a069887.jpg">Plot of a(n)/(n*log(n)) for n = 2..10000</a>
%F A069887 Asymptotically it seems that a(n) ~ C*n*log(n) where C = 0.84... is close to the constant described in A055573.
%e A069887 The simple continued fraction for (1+1/10)^10 is [2, 1, 1, 2, 5, 1, 128, 1, 2, 12, 5, 3, 46, 1, 11, 7] which contains 16 elements, hence a(10) = 16.
%t A069887 Table[Length[ContinuedFraction[(1+1/n)^n]],{n,70}] (* _Harvey P. Dale_, Jun 12 2013 *)
%Y A069887 Cf. A001113, A055573, A069655.
%K A069887 easy,nonn
%O A069887 1,2
%A A069887 _Benoit Cloitre_, May 04 2002