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Continued Fractions with Non-Integer Numerators
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John Greene and Jesse Schmieg

Department of Mathematics and Statistics

University of Minnesota Duluth

Duluth, MN 55812

USA

**Abstract:**

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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