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I encountered the recursion $$\frac{a[n+2]}{a[n+1]}-e^{-\frac{a[n+1]}{a[n]}}=0$$ when trying to explain why points are apparently arranged along an exponential curve in the scatter plot of an empiric dataset depicted in the image below.

I am however interested in methods for solving the recursion itself, or more generally recursions with a functional dependency of $\frac{a[n+2]}{a[n+1]}$ on $\frac{a[n+1]}{a[n]}$

I know that generating functions may help, but would appreciate concrete steps for their application to the above recursion.

Scatter plot with hidden exponential

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  • $\begingroup$ If you look at $b(n) = \frac{a(n+1)}{a(n)}$ you get $b(n) = \exp_{1/e}^n(b(0))$ — tetration. $\endgroup$
    – Daniel Weber
    Commented Apr 17 at 6:49
  • $\begingroup$ This should converge quite quickly to the omega constant $\Omega$ (which is the fixed point of $e^{-x}$), so you can expect $a(n)$ to be proportional to $\Omega^n$ $\endgroup$
    – Daniel Weber
    Commented Apr 17 at 6:52
  • $\begingroup$ @CommandMaster could you please elaborate a bit on the tetration expression in your first comment, resp. where can I find an explanation of the notation? $\endgroup$
    – Manfred Weis
    Commented Apr 17 at 6:59
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    $\begingroup$ $b(n)$ satisfies $b(n+1)=(1/e)^{b(n)}$, so $b(n) = (1/e)^{(1/e)^{\dots^{b(0)}}}$ with $n$ exponentiations. This is exactly tetration, which you can read about on Wikipedia $\endgroup$
    – Daniel Weber
    Commented Apr 17 at 7:17
  • $\begingroup$ Please state what the axes of the plot represent. $\endgroup$
    – Daniel Asimov
    Commented Apr 17 at 23:59

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