I've recently come across a sequence of numbers but I can't determine any sensible way to say what the $I^{th}$ number in the pattern will be. The pattern is as follows:
$\frac{1}{2}, \\\frac{1}{4}, \frac{3}{4}, \\\frac{1}{8}, \frac{5}{8}, \frac{3}{8}, \frac{7}{8}, \\\frac{1}{16}, \frac{9}{16}, \frac{5}{16}, \frac{13}{16}, \frac{3}{16}, \frac{11}{16}, \frac{7}{16}, \frac{15}{16},\\ \frac{1}{32}, \frac{17}{32}, \frac{9}{32}, \frac{25}{32}, \frac{5}{32}, \frac{21}{32}, \frac{13}{32}, \frac{29}{32}, \frac{3}{32}, \frac{19}{32}, \frac{11}{32}, \frac{27}{32}, \frac{7}{32}, \frac{23}{32}, \frac{15}{32}, \frac{31}{32}, \\ \textrm{and so on. }$
The denominator is easy to determine, and it's also clear that once a numerator appears, it will also be in subsequent levels, but how the "new numerators" at each level are determined is unclear to me. If I find this I may be able to determine an expression for the $I^{th}$ number.
Ignore the denominator. For the $i^\text{th}$ number, convert that number to binary, reverse the order of the digits, and then convert that back to a base 10 number.