# Interesting Recursive Continued Fraction Limit

I was messing around with recursive functions the other day and came up with something that could be interesting: Definition of $\bar{\Xi}(n)$:\ Let $\Xi (n)=\frac{2n}{\Xi(2n)}+\frac{2n+1}{\Xi(2n+1)}$ Then $\bar{\Xi}(n)=\frac{n}{\Xi(n)}=\frac{n}{\frac{2n}{\Xi(2n)}+\frac{2n+1}{\Xi(2n+1)}}$ Starting with $n=1$ we have: \ $\frac{1}{\frac{2}{\frac{4}{\frac{8}{\frac{16}{32+33}+\frac{17}{34+35}}+\frac{9}{\frac{18}{36+37}+\frac{19}{38+39}}}+\frac{5}{\frac{10}{\frac{20}{40+41}+\frac{21}{42+43}}+\frac{11}{\frac{22}{44+45}+\frac{23}{46+47} }}}+\frac{3}{\frac{6}{\frac{12}{\frac{24}{48+49}+\frac{25}{50+51}}+\frac{13}{\frac{26}{52+53}+\frac{27}{54+55}}}+\frac{7}{\frac{14}{\frac{28}{56+57}+\frac{29}{58+59}}+\frac{15}{\frac{30}{60+61}+\frac{31}{62+63}}}}}$

My question is how i would go about evaluating this fraction starting with n=1 and iterating the recursion infinitely many times. There are also a few interesting patterns in the fractions, If you find any good ones let me know. Just curious about the nature of this type of function, not really looking for an exact answer, just wondering if it has a classification or a known behavior. Thanks!

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