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After reading some articles about how Archimedes used to approximate $\pi$ and doing some math, I found out that the following formula spits out the length of the circumference $C(R)$ of a circle of radius $R$ (here $a_n=\sqrt{2-\sqrt{4-a^2_{n-1}}}$ with $a_1=1$): $$C(R)=3R\lim_{n\rightarrow\infty}{(2^n a_{n-1})}$$ (The awesome thing is that it doesn't involve $\pi$ at all, so it can actually be used to prove that $\pi$ is in fact constant.) Are there any other explicit formulas for the length of the circumference of a circle of radius $R$ that don't involve $\pi$ in any way? It would be awesome if they involve $e$, $i$ or $\phi$ or some other neat constants in a nontrivial way (so don't post the formula above with $\frac{e}{e}$ outfront and things like that).

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    $\begingroup$ You have given a starting point for the relation with $a_1 = 1$ but have not defined any sort of recurrence, eg $a_n = f(a_{n-1})$ so it doesn't make sense. $\endgroup$ – user157545 Jun 24 '14 at 11:06
  • $\begingroup$ @user157545 my bad, fixed. $\endgroup$ – user132181 Jun 24 '14 at 11:09
  • $\begingroup$ There are huge numbers of formulae of this type. For a start try Mathworld. $\endgroup$ – David Jun 24 '14 at 11:11
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    $\begingroup$ $\pi=-i\cdot\ln(-1)$ $\endgroup$ – Lucian Jun 24 '14 at 13:13

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