Search: id:A003417
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%I A003417 M0088
%S A003417 2,1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,1,1,14,1,1,16,1,1,18,1,1,
%T A003417 20,1,1,22,1,1,24,1,1,26,1,1,28,1,1,30,1,1,32,1,1,34,1,1,36,1,1,38,1,1,
               40,1,1,42,
%U A003417 1,1,44,1,1,46,1,1,48,1,1,50,1,1,52,1,1,54,1,1,56,1,1,58,1,1,60,1,1,62,
               1,1,64,1,1,66
%N A003417 Continued fraction for e.
%C A003417 This is also the Engel expansion for 3*exp(1/2)/2 - 1/2. - Gerald McGarvey 
               (Gerald.McGarvey(AT)comcast.net), Aug 07 2004
%C A003417 First differences are A120691. - Paul Barry (pbarry(AT)wit.ie), Jun 27 
               2006
%D A003417 H. Cohn, A short proof of the simple continued fraction expansion of 
               e, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62.
%D A003417 CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
%D A003417 S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.2.
%D A003417 J. R. Goldman, The Queen of Mathematics, 1998, p. 70.
%D A003417 T. J. Osler, A proof of the continued fraction expansion of e^(1/M), 
               Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
%D A003417 O. Perron, Die Lehre von den Kettenbr\"{u}chen, 2nd ed., Teubner, Leipzig, 
               1929, p. 134.
%D A003417 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A003417 N. J. A. Sloane, <a href="b003417.txt">Table of n, a(n) for n = 1..10000</
               a>
%H A003417 K. Matthews, <a href="http://www.numbertheory.org/php/davison.html">Finding 
               the continued fraction of e^(l/m)</a>
%H A003417 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A003417 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%H A003417 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               e.html">Link to a section of The World of Mathematics.</a>
%H A003417 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~number~contfrac.en.html">
               Contfrac</a>
%H A003417 <a href="Sindx_Con.html#confC">Index entries for continued fractions 
               for constants</a>
%F A003417 G.f.: (2+x+2x^2-3x^3-x^4+x^6)/(1-2x^3+x^6); a(n)=sum{k=0..n, 2*C(0,k)-C(1,
               k)-2*sin(2*pi*(k-1)/3)*floor((2k-1)/3)/sqrt(3)} [offset 0]; - Paul 
               Barry (pbarry(AT)wit.ie), Jun 27 2006
%F A003417 a(n)=2*a(n-3)-a(n-6), n>=8 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Feb 10 2009]
%p A003417 numtheory[cfrac](exp(1),100,'quotients'); - Jani Melik (jani_melik(AT)hotmail.com), 
               May 25 2006
%p A003417 A003417:=(2+z+2*z**2-3*z**3-z**4+z**6)/(z-1)**2/(z**2+z+1)**2; [S. Plouffe 
               in his 1992 dissertation.]
%t A003417 ContinuedFraction[E, 100] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               Apr 07 2006
%o A003417 (PARI) contfrac(exp(1)-1) - Alexander R. Povolotsky (pevnev(AT)juno.com), 
               Feb 23 2008
%o A003417 (PARI) { allocatemem(932245000); default(realprecision, 25000); x=contfrac(exp(1)); 
               for (n=1, 10000, write("b003417.txt", n, " ", x[n])); } [From Harry 
               J. Smith (hjsmithh(AT)sbcglobal.net), Apr 14 2009]
%Y A003417 Cf. A001113, A007676, A007677.
%Y A003417 Sequence in context: A078997 A024680 A083531 this_sequence A158986 A079900 
               A117354
%Y A003417 Adjacent sequences: A003414 A003415 A003416 this_sequence A003418 A003419 
               A003420
%K A003417 nonn,cofr,nice,easy
%O A003417 1,1
%A A003417 N. J. A. Sloane (njas(AT)research.att.com).

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