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A005131 A generalized continued fraction for Euler's number e. 7
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; text; internal format)
OFFSET

0,5

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, Apr 19 2011

REFERENCES

Douglas Hofstadter, "Fluid Concepts and Creative Analogies: Computer Models of the Fundamental Mechanisms of Thought".

LINKS

N. J. A. Sloane, Table of n, a(n) for n = 0..5000

H. Cohn, A short proof of the simple continued fraction expansion of e, Amer. Math. Monthly, 113 (No. 1, 2006), 57-62. [JSTOR] and arXiv:math/0601660.

D. H. Lehmer, Continued fractions containing arithmetic progressions, Scripta Mathematica, 29 (1973): 17-24. [Annotated copy of offprint]

T. J. Osler, A proof of the continued fraction expansion of e^(1/M), Amer. Math. Monthly, 113 (No. 1, 2006), 62-66.

A. J. van der Poorten, Continued fraction expansions of values of the exponential function...

A. J. van der Poorten, Number theory and Kustaa Inkeri

FORMULA

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

G.f. = (-x^5 + 2*x^4 - x^3 + x^2 + 1)/(x^6 - 2*x^3 + 1). - Alexander R. Povolotsky, 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

a(n) = 1 + 2*(2*n-5) * (cos(2*Pi*(n-1)/3)+1/2)/9. - David Spitzer, Jan 09 2017

MATHEMATICA

Table[If[Mod[k, 3] == 1, 2/3*(k - 1), 1], {k, 0, 80}] (* Joseph Biberstine (jrbibers(AT)indiana.edu), Aug 14 2006 *)

PROG

(PARI) a(n)=if(n>=0, [1, 2*(n\3), 1][n%3+1]) \\ [From Jaume Oliver Lafont, Nov 14 2009]

CROSSREFS

Cf. A003417, A100261.

Sequence in context: A061462 A122578 A208648 * A105477 A127709 A226174

Adjacent sequences:  A005128 A005129 A005130 * A005132 A005133 A005134

KEYWORD

nonn,cofr

AUTHOR

Russ Cox

EXTENSIONS

Edited by M. F. Hasler, Jan 26 2014

STATUS

approved

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Last modified March 27 08:32 EDT 2017. Contains 284146 sequences.