\[\begin{array}{l} \frac{e}{2} = 1 + \frac{1}{{2 + \frac{1}{{1 + \frac{1}{{2 + \frac{2}{{1 + \frac{1}{{2 + \frac{3}{{1 + ...}}}}}}}}}}}} \hfill \\ {b_i} = \left[ {1,1,1,2,1,3,1,4,1,5,1,...} \right] \hfill \\ \end{array}\]