What is the fastest way to $\pi$? There are many known sequences convergent to $\pi$ with different convergence accelerations. For example both of the following sequences are convergent to $\pi$ when $n$ goes to $\infty$:
(a) $a_n=2^{n+1}\times\sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{...+\sqrt{2}}}}}}}$ which the under brace part repeats $n$ times. More clear:
$a_0=2\sqrt{2}$
$a_1=4\sqrt{2-\sqrt{2}}$
$a_2=8\sqrt{2-\sqrt{2+\sqrt{2}}}$
$a_3=16\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}$  
$...$
(b) $b_n=4\times\sum_{i=0}^{n}\frac{(-1)^i}{2i+1}$ 
More clear:
$b_0=4$
$b_1=4(1-\frac{1}{3})$
$b_2=4(1-\frac{1}{3}+\frac{1}{5})$
$b_3=4(1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7})$
$...$
It seems the first sequence goes to $\pi$ faster than the second one.
Question: What is the fastest known explicit sequence convergent to $\pi$? Please introduce a list of known sequences convergent to $\pi$ and their convergence accelerations.

The provably fastest convergence, in fractions, is given by the continued fraction: 
$$[3;7,15,1,292,1,1,1,2,1,3,1,\ldots]$$
This corresponds to the sequence $$3,\frac{22}{7}, \frac{333}{106}, \frac{355}{113}, \frac{103993}{33102},\ldots$$
