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A003417
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Continued fraction for e.
(Formerly M0088)
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12
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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, 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, 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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| This is also the Engel expansion for 3*exp(1/2)/2 - 1/2. - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Aug 07 2004
First differences are A120691. - Paul Barry (pbarry(AT)wit.ie), Jun 27 2006
Sorted with duplicate terms dropped, this is A004277, 1 together with the positive even numbers. - Alonso del Arte, Jan 27 2012
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REFERENCES
| H. Cohn, A short proof of the simple continued fraction expansion of e, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.3.2.
J. R. Goldman, The Queen of Mathematics, 1998, p. 70.
T. J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
O. Perron, Die Lehre von den Kettenbr\"{u}chen, 2nd ed., Teubner, Leipzig, 1929, p. 134.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 1..10000
K. Matthews, Finding the continued fraction of e^(l/m)
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
G. Xiao, Contfrac
Index entries for continued fractions for constants
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FORMULA
| 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
a(n)=2*a(n-3)-a(n-6), n>=8 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 10 2009]
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EXAMPLE
| 2.718281828459... = 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + ...))))
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MAPLE
| numtheory[cfrac](exp(1), 100, 'quotients'); # Jani Melik (jani_melik(AT)hotmail.com), May 25 2006
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.
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MATHEMATICA
| ContinuedFraction[E, 100] (* Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Apr 07 2006 *)
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PROG
| (PARI) contfrac(exp(1)-1) \\ Alexander R. Povolotsky (pevnev(AT)juno.com), Feb 23 2008
(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
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CROSSREFS
| Cf. A001113, A007676, A007677.
Sequence in context: A078997 A024680 A083531 * A158986 A079900 A188317
Adjacent sequences: A003414 A003415 A003416 * A003418 A003419 A003420
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KEYWORD
| nonn,cofr,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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