A005131 A generalized continued fraction for Euler's number e.

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

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..500

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]) \\ Jaume Oliver Lafont, Nov 14 2009

Crossrefs

Cf. A003417, A100261.

Sequence in context: A294334 A122578 A208648 * A105477 A325772 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