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We can see this is related by \begin{equation}  \frac{N[\cot(x)]}{N[tan(x)]}=1,1,1,\frac{1}{17},\frac{1}{31},1,\frac{1}{5461},\frac{1}{257},\frac{1}{73},\frac{1}{1271}  \end{equation}  Where $N[f(x)]$ is the numerator of a Taylor expanded $f$. However there is one later coefficient which doesn't seem to agree. $4097$ in the original sequence has in this cot/tan expansion $241$ which is $17$ times too small. Odd that $17$ is the first coefficient,