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Geometry Level 4

Tn=cot12+cot18+cot118+cot12n2\mathfrak{T}_n=\cot^{-1} 2+\cot^{-1} 8+\cot^{-1} 18+\cdots \cot^{-1} 2n^2

limnTn=απγ\displaystyle \lim_{n\to\infty}\mathfrak{T}_n=\dfrac{\color{#D61F06}{\alpha }\pi}{\color{#3D99F6}{\gamma}} where α\color{#D61F06}{\alpha} and γ\color{#3D99F6}{\gamma} are coprime. α+4γ= ?\Large \color{#D61F06}{\alpha} +4\color{#3D99F6}{\gamma}=\ ?

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