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Find a closed form for this sequence: $a_{n+1} = a_n + a_n^{-1}$

Suppose $a_{n+1}=a_n+\frac{1}{a_n}$ where $a_{1}=1$. Is $f(n)=a_n$ an elementary function?

I haven't found any paper concerning it. Thanks for your attention!

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    I know some of the closed form such as $\sqrt{2n+\frac{\log{n}}{2}}$, but I need the elementariness of function $f$, not the upper/lower bound or equivalent infinite2012-08-20

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