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I have The following non linear recursive sequence:

$R_{n+1}=R_{n}+\frac{1}{R_n}$, where $R_1=1$.

How do I find a closed-form function for this sequence, is it possible?

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I don't think that this has a non-recursive closed form, but it is the same as the fractional chromatic number of the of the Mycielski graph. See A073833 and A073834 on the OEIS for the numerators and denominators of this sequence, respectively.

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May help you: I tried a Matlab program

clc;  

clear;  
r=[];  

r(1)=1;

N=1000;

for n=2:N

    r(n)=r(n-1)+1/r(n-1)

end;

Put $N$ various number, $r_{1000}=44.7569$. It seems to be divergent...

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  • $\begingroup$ While you are it it: investigate $r(n)-\sqrt{2n}$. No, I don't think there is a closed form, but the growth is rather regular and predictable. $\endgroup$ – Professor Vector Sep 28 '17 at 19:12
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    $\begingroup$ It doesn’t matter if the sequence grows unbounded. $\endgroup$ – MrYouMath Sep 28 '17 at 19:19

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