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I try to solve the following recursion for $n \in \mathbb{N}$.

$r_i = r_{i-1} - \frac{1}{2} \cdot \sqrt{1 - \frac{4\pi^2\cdot r_{i-1}^2}{n^2} \cdot \cos^2 \left(\frac{\pi}{n}\right)}$

$r_0 = \frac{n}{2\pi}$

I translated it into the following code for Mathematica:

RSolve[{g[x]==g[x-1]- 1/2 Sqrt[1 - 4 Pi^2  g[x-1]^2/n^2  (Cos[Pi/n])^2], g[0]==n/(2  Pi)}, g[x], x]

However, Mathematica cannot interpret this and I get the input as result. Is there any mistake from my side or is Mathematica not able to solve this?

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    $\begingroup$ Almost certainly the latter. $\endgroup$ – Daniel Lichtblau Mar 14 at 14:22
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Will using RecurrenceTable helps?

RecurrenceTable[{g[x] ==  g[x - 1] - 1/2 Sqrt[1 - 4 Pi^2 g[x - 1]^2/n^2 (Cos[Pi/n])^2], 
      g[0] == n/(2 Pi)}, g, {x, 1, 3}]
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