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
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
Programs
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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 *) -
PARI
a(n)=if(n>=0,[1, 2*(n\3), 1][n%3+1]) \\ Jaume Oliver Lafont, Nov 14 2009
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
Extensions
Edited by M. F. Hasler, Jan 26 2014
Comments