<s>[INST] A couple of formulas for $\pi$ While I was studying sums of polygonal numbers I discovered a couple of formulas for $\pi$. Most of the formulas I found were already known, but I can't seem to find any references to the following four:
$$\pi=\sum_{n=1}^\infty \frac{3}{n(2n-1)(4n-3)},$$
$$\pi=\sum_{n=1}^\infty \frac{3\sqrt{3}}{(3n-1)(3n-2)},$$
$$\pi=4\sqrt{3}\sum_{n=1}^\infty \frac{12n-5}{8n(2n-1)(3n-1)(6n-5)},$$
$$\pi=16\sum_{n=1}^\infty \frac{864n(n-1)+226}{(12n-1)(12n-5)(12n-7)(12n-11)(4n-1)(4n-3)}.$$
Are the above formulas known? I have looked here and here. Because of their simplicity, I find it hard to believe that the first two formulas are unknown.
 [/INST] The first formula may be rewritten by changing the lower limit of the summation:
$$\begin{align}
\frac{\pi}{3}
&=\sum_{n=1}^\infty \frac{1}{n(2n-1)(4n-3)} \\
&=\sum_{n=0}^\infty \frac{1}{(n+1)(2(n+1)-1)(4(n+1)-3)} \\
&=\sum_{n=0}^\infty \frac{1}{(n+1)(2n+1)(4n+1)} \\
\end{align}$$
This last equation is given by Lehmer in page 139 here.
A similar one that uses four factors is
$$\pi-3=\sum_{k=0}^\infty \frac{24}{(4k+2)(4k+3)(4k+5)(4k+6)}$$
(see Series and integrals for inequalities and approximations to $\pi$)
A formula which is similar to the third one you give is:
$$\pi=72 \sqrt{3} \sum_{k=0}^\infty \frac{1}{(6 k+1) (6 k+2) (6 k+4) (6 k+5)}$$
which has all terms positive and constant numerator. Taking the first term of the series out of the summation, this proves $\pi>\frac{9}{5}\sqrt{3}\approx 3.12$.
There is also
$$\pi=\frac{23040(17+9 \sqrt{3})}{23}  \sum_{k=0}^\infty \frac{1}{(12 k+1) (12 k+3) (12 k+5) (12 k+7) (12 k+9) (12 k+11)}$$
with the denominators of your fourth formula. The first term in this series leads to the approximation
$$\pi \approx \frac{2^9(17+9\sqrt{3})}{5313}=\frac{2^{10}((2^4+1)+(2^3+1)\sqrt{3})}{21·22·23} = \frac{2^{10}}{231(17-9\sqrt{3})}\approx 3.140$$
Using $163$ instead of $\frac{5313}{17+9\sqrt{3}} \approx 163.033$ gets one more correct digit. The approximation
$$\pi \approx \frac{2^9}{163} \approx 3.1411$$ 
is due to Stoschek (see http://mathworld.wolfram.com/PiApproximations.html)
Also compare to $\pi \approx \frac{2^{13}}{3(-383+560\sqrt{5})}$ given by T.Piezas here.
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