Table of Contents
ISRN Combinatorics
Volume 2013, Article ID 414623, 5 pages
http://dx.doi.org/10.1155/2013/414623
Research Article

Closed Form Continued Fraction Expansions of Special Quadratic Irrationals

1Goldman, Sachs & Co., 85 Broad Street, New York, NY 10004, USA
2Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, USA

Received 12 November 2012; Accepted 28 November 2012

Academic Editors: C. da Fonseca, D. S. Kim, and W. Menasco

Copyright © 2013 Daniel Fishman and Steven J. Miller. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We derive closed form expressions for the continued fractions of powers of certain quadratic surds. Specifically, consider the recurrence relation with , , a positive integer, and (note that gives the Fibonacci numbers). Let . We find simple closed form continued fraction expansions for for any integer by exploiting elementary properties of the recurrence relation and continued fractions.