%I A005131
%S A005131 1,0,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,14,1,1,16,1,1,18,
%T A005131 1,1,20,1,1,22,1,1,24,1,1,26,1,1,28,1,1,30,1,1,32,1,1,34,1,1,36,1,1,
%U A005131 38,1,1,40,1,1,42
%N A005131 A generalized continued fraction for Euler's number e.
%C A005131 Only a(1) = 0 prevents this from being a simple continued fraction. The 
               motivation for this alternate representation is that the simple pattern 
               {1, 2*n, 1} (from n=0) may be more mathematically appealing than 
               the pattern in the corresponding simple continued fraction (at A003417) 
               - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006
%D A005131 H. Cohn, A short proof of the simple continued fraction expansion of 
               e, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62.
%D A005131 Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer 
               Models of the Fundamental Mechanisms of Thought".
%D A005131 T. J. Osler, A proof of the continued fraction expansion of e^(1/M), 
               Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
%H A005131 N. J. A. Sloane, <a href="b005131.txt">Table of n, a(n) for n = 0..5000</
               a>
%H A005131 A. J. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/alfpapers/
               a117.pdf">Continued fraction expansions of values of the exponential 
               function...</a>
%H A005131 A. J. van der Poorten, <a href="http://www-centre.mpce.mq.edu.au/alfpapers/
               a151.pdf">Number theory and Kustaa Inkeri</a>
%F A005131 If Mod[n,3]==1, a(n) = 2*(k-1)/3, else a(n) = 1. - Joseph Biberstine 
               (jrbibers(AT)indiana.edu), Aug 14 2006
%F A005131 G.f. = (-x^5 + 2*x^4 - x^3 + x^2 + 1)/(x^6 - 2*x^3 + 1) - Alexander R. 
               Povolotsky (pevnev(AT)juno.com), Apr 26 2008
%F A005131 {-a(n)-2*a(n+1)-3*a(n+2)-2*a(n+3)-a(n+4)+2*n+8, a(0) = 1, a(1) = 0, a(2) 
               = 1, a(3) = 1, a(4) = 2, a(5) = 1}. - Robert Israel, May 14 2008
%t A005131 Table[If[Mod[k, 3] == 1, 2/3*(k - 1), 1], {k, 0, 80}] - Joseph Biberstine 
               (jrbibers(AT)indiana.edu), Aug 14 2006
%o A005131 (PARI) a(n)=if(n>=0,[1, 2*(n\3), 1][n%3+1]) [From Jaume Oliver Lafont 
               (joliverlafont(AT)gmail.com), Nov 14 2009]
%Y A005131 Cf. A003417, A100261.
%Y A005131 Sequence in context: A141450 A061462 A122578 this_sequence A105477 A127709 
               A131350
%Y A005131 Adjacent sequences: A005128 A005129 A005130 this_sequence A005132 A005133 
               A005134
%K A005131 nonn,cofr,new
%O A005131 0,5
%A A005131 Russ Cox (rsc(AT)swtch.com)