Algorithms of transformation between positive and even continued fractions (Q6611872)

In this paper, the author considers a finite continued fraction of the form \N\[\N [x_1,x_2,\dots,x_{N-1},x_N]:=x_1+\cfrac{1}{x_2+\cfrac{1}{\ddots+x_{N-1}+\frac{1}{x_N}}}. \N\]\NIf all \(x_i\) (\(i=1,2,\dots,N\)) are positive integers, the author calls it a (finite) positive continued fraction. If all \(x_i\) (\(i=1,2,\dots,N\)) are positive or negative even integers, the author calls it a (finite) even continued fraction. Then, he gives the algorithm transforming positive continued fractions into even continued fractions, and the algorithm transforming even continued fractions into positive continued fractions. These algorithms can be applied to infinite continued fractions and the even continued fraction expansions of \(e\) and \(\sqrt{e}\) are given.