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A003417
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Continued fraction for e.
(Formerly M0088)
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19
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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
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OFFSET
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1,1
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COMMENTS
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This is also the Engel expansion for 3*exp(1/2)/2 - 1/2. - Gerald McGarvey, Aug 07 2004
First differences are A120691. - Paul Barry, 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
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H. Cohn, A short proof of the simple continued fraction expansion of e, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62. Available at http://www.jstor.org/stable/27641837 and arXiv:math/0601660.
CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 88.
S. Crowley, Mellin and Laplace Integral Transforms Related to the Harmonic Sawtooth Map and a Diversion Into The Theory Of Fractal Strings, http://vixra.org/pdf/1202.0079v2.pdf, 2012.
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).
Ward. O. Whitt, Weirdness in CTMC's, Notes for Course IEOR 6711: Stochastic Models I, http://www.columbia.edu/~ww2040/6711F12/lect1129.pdf, 2012. - From N. J. A. Sloane, Jan 03 2013
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 1..10000
K. Matthews, Finding the continued fraction of e^(l/m)
_Simon 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.
_Simon Plouffe_, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, e.
G. Xiao, Contfrac
Index entries for continued fractions for constants
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FORMULA
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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, Jun 27 2006
a(n)=2*a(n-3)-a(n-6), n>=8 . [From Philippe DELEHAM, Feb 10 2009]
G.f.: 1 + U(0) where U(k)= 1 + x/(1 - x*(2*k+1)/(1+x*(2*k+1) - 1/((2*k+1) + 1 - (2*k+1)*x/(x + 1/U(k+1))))) ; (continued fraction, 5-step). - Sergei N. Gladkovskii, Oct 07 2012
a(3n)=2n, a(1)=2, a(n)=1 else (i.e., for n>1, not multiple of 3). - M. F. Hasler, May 01 2013
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EXAMPLE
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2.718281828459... = 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + ...))))
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MAPLE
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numtheory[cfrac](exp(1), 100, 'quotients'); # Jani Melik, May 25 2006
A003417:=(2+z+2*z**2-3*z**3-z**4+z**6)/(z-1)**2/(z**2+z+1)**2; # Simon Plouffe in his 1992 dissertation.
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MATHEMATICA
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ContinuedFraction[E, 100] (* Stefan Steinerberger, Apr 07 2006 *)
a[n_] := KroneckerDelta[1, n] + 2n/3 - (2n-3)/3*DirichletCharacter[3, 1, n]; Table[a[n], {n, 1, 20}] (* Enrique Pérez Herrero, Feb 23 2013 *)
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PROG
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(PARI) contfrac(exp(1)-1) \\ Alexander R. Povolotsky, 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, Apr 14 2009
(PARI) A003417(n)=if(n%3, 1+(n==1), n\3*2) \\ [M. F. Hasler, May 01 2013]
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CROSSREFS
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Cf. A001113, A007676, A007677.
Cf. A001204, A058282.
Cf. A005131.
Sequence in context: A078997 A024680 A083531 * A158986 A079900 A188317
Adjacent sequences: A003414 A003415 A003416 * A003418 A003419 A003420
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KEYWORD
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nonn,cofr,nice,easy
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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