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A sequence is defined by: $$a_{n+1}=a_n+\frac{1}{a_n}$$ With $a_1=1$

Is a closed-form expression for this sequence possible? A search turns up the rule that all expressions with linear recurrence and constant coefficients will definitely have a closed-form representation. But this is not a linear expression, since i's definition includes the reciprocal of the previous term.

If it is possible, what could it be?

Arthur
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Siddhartha
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  • WA doesn't find a closed down. – Arthur Oct 26 '14 at 13:30
  • I cannot provide evidence, but I seriously doubt there is a close form expression for this. – Asier Calbet Oct 26 '14 at 13:30
  • If you interpret $a_n$ as a continued fraction then $a_{n + 1}$ is the sum of two continued fractions of known coefficients. So it may just be possible to find a closed form using continued fraction arithmetic, but I doubt it. Maybe someone knows a trick for this one though. – Thomas Oct 26 '14 at 15:42

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