recurrence relations - Closed form for the sequence defined by $a_0=1$ and $a_{n+1} = a_n + a_n^{-1}$

Today, we had a math class, where we had to show, that $a_{100} > 14$ for

$$a_0 = 1;\qquad a_{n+1} = a_n + a_n^{-1}$$

Apart from this task, I asked myself: Is there a closed form for this sequence? Since I didn't find an answer by myself, can somebody tell me, whether such a closed form exists, and if yes what it is?

Answer

I agree, a closed form is very unlikely.
As for more precise asymptotics, I think $a_n = \sqrt{2n} + 1/8\,{\frac {\sqrt {2}\ln \left( n \right) }{\sqrt {n}}}-{\frac {1}{
128}}\,{\frac {\sqrt {2} \left( \ln \left( n \right) -2 \right) ^{2} + o(1)}
{{n}^{3/2}}}$