Journal of Integer Sequences, Vol. 20 (2017), Article 17.1.2

Continued Fractions with Non-Integer Numerators

John Greene and Jesse Schmieg
Department of Mathematics and Statistics
University of Minnesota Duluth
Duluth, MN 55812


Anselm and Weintraub investigated a generalization of classic continued fractions, where the "numerator" 1 is replaced by an arbitrary positive integer. Here, we generalize further to the case of an arbitrary real number z ≥ 1. We focus mostly on the case where z is rational but not an integer. Extensive attention is given to periodic expansions and expansions for √n, where we note similarities and differences between the case where z is an integer and when z is rational. When z is not an integer, it need no longer be the case that √n has a periodic expansion. We give several infinite families where periodic expansions of various types exist.

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Received September 18 2016; revised version received December 21 2016. Published in Journal of Integer Sequences, December 23 2016.

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