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A005131
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A generalized continued fraction for Euler's number e.
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3
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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, 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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| 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
If we consider a(n) = A005131(n+1), n >= 0, then we get the simple continued fraction for 1/(e-1). - Daniel Forgues, April 19, 2011
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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.
Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought".
T. J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 0..5000
A. J. van der Poorten, Continued fraction expansions of values of the exponential function...
A. J. van der Poorten, Number theory and Kustaa Inkeri
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FORMULA
| If Mod[n,3]==1, a(n) = 2*(k-1)/3, else a(n) = 1. - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006
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
{-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
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MATHEMATICA
| Table[If[Mod[k, 3] == 1, 2/3*(k - 1), 1], {k, 0, 80}] - Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006
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PROG
| (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]
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CROSSREFS
| Cf. A003417, A100261.
Sequence in context: A141450 A061462 A122578 * A105477 A127709 A199856
Adjacent sequences: A005128 A005129 A005130 * A005132 A005133 A005134
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KEYWORD
| nonn,cofr
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AUTHOR
| Russ Cox (rsc(AT)swtch.com)
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