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%I #47 Feb 21 2018 03:30:14
%S 1,0,1,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,14,1,1,16,1,1,18,1,1,
%T 20,1,1,22,1,1,24,1,1,26,1,1,28,1,1,30,1,1,32,1,1,34,1,1,36,1,1,38,1,
%U 1,40,1,1,42
%N A generalized continued fraction for Euler's number e.
%C 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
%C If we consider a(n) = A005131(n+1), n >= 0, then we get the simple continued fraction for 1/(e-1). - _Daniel Forgues_, Apr 19 2011
%D Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought".
%H N. J. A. Sloane, <a href="/A005131/b005131.txt">Table of n, a(n) for n = 0..500</a>
%H H. Cohn, A short proof of the simple continued fraction expansion of e, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62. <a href="http://www.jstor.org/stable/27641837">[JSTOR]</a> and <a href="http://arxiv.org/abs/0601660">arXiv:math/0601660</a>.
%H D. H. Lehmer, <a href="/A016825/a016825.pdf">Continued fractions containing arithmetic progressions</a>, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]
%H T. J. Osler, <a href="http://www.jstor.org/stable/27641838">A proof of the continued fraction expansion of e^(1/M)</a>, Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
%H A. J. van der Poorten, <a href="http://web.archive.org/http://www-centre.mpce.mq.edu.au/alfpapers/a117.pdf">Continued fraction expansions of values of the exponential function...</a>
%H A. J. van der Poorten, <a href="http://web.archive.org/web/20040405083745/http://www-centre.mpce.mq.edu.au/alfpapers/a151.pdf">Number theory and Kustaa Inkeri</a>
%F If n==1 (mod 3), then a(n) = 2*(n-1)/3, otherwise a(n) = 1. - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006
%F G.f. = (-x^5 + 2*x^4 - x^3 + x^2 + 1)/(x^6 - 2*x^3 + 1). - _Alexander R. Povolotsky_, Apr 26 2008
%F {-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
%F a(n) = 1 + 2*(2*n-5) * (cos(2*Pi*(n-1)/3)+1/2)/9. - _David Spitzer_, Jan 09 2017
%t Table[If[Mod[k, 3] == 1, 2/3*(k - 1), 1], {k, 0, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)
%o (PARI) a(n)=if(n>=0,[1, 2*(n\3), 1][n%3+1]) \\ _Jaume Oliver Lafont_, Nov 14 2009
%Y Cf. A003417, A100261.
%K nonn,cofr
%O 0,5
%A _Russ Cox_
%E Edited by _M. F. Hasler_, Jan 26 2014
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