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The following sequence appeared in IMC 2012 (a math competition): $a_1 = \frac{1}{2}, \qquad a_{n+1} = \frac{n a_n^2}{1+(n+1)a_n}$

I am trying to find an explicit formula for the sequence. It seems to be nicer to look at $b_n = a_n^{-1}$: $b_1 = 2, \qquad b_{n+1} = \frac{b_n(b_n +n +1)}{n}$ Can some closed formula be derived?

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    On purely empirical grounds, I would argue that no closed formula can be found. The point is, this sequence was stared at rather hard by a lot of contestants and team leaders. If anybody had found a closed formula for $a_n$, then this would probably make its way into the solution, and would surely be quickly spread after the contest via small talk. Since neither of these was the case, I suspect that closed formula either does not exist, or is very complex/hard to find.2013-04-28

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Numerically

$ b(n) \sim n c^{2^n} $ where c = 1.36534926036757464312824443040683531215776134381623126072...

The asymptotic ratio

I created also a new sequence in OEIS, see http://oeis.org/A258622