<s>[INST] $2\arctan(\phi^{-n})=\arctan\frac{p}{q}$ or $\arctan\frac{p\sqrt{5}}{q}$, where $\phi$ is the Golden Ratio. Is there a pattern in the $\frac{p}{q}$s? It is very interesting to know that 
$$\arctan\frac{1}{\phi} + \arctan\frac{1}{\phi^3}= \arctan 1 = \frac{\pi}{4}$$
where Golden ratio $\phi = \frac12(\sqrt5 +1)$ is in association with circle constant $\pi$.
More interesting phenomenon is evaluation of inverse tan functions of inverse of $\phi$ in its consecutive powers as follows
$$\begin{align}
2\arctan\frac{1}{\phi} &= \arctan 2 &
2\arctan\frac{1}{\phi^2} &= \arctan\frac{2\sqrt{5}}{5}\\
2\arctan\frac{1}{\phi^3} &= \arctan\frac{1}{2} &
2\arctan\frac{1}{\phi^4} &= \arctan\frac{2\sqrt{5}}{15}\\
2\arctan\frac{1}{\phi^5} &= \arctan\frac{2}{11} &
2\arctan\frac{1}{\phi^6} &= \arctan\frac{\sqrt5}{20} \\
2\arctan\frac{1}{\phi^7} &= \arctan\frac{2}{29} &
2\arctan\frac{1}{\phi^8} &= \arctan\frac{2\sqrt5}{105} \\
2\arctan\frac{1}{\phi^9} &= \arctan\frac{1}{38} &
2\arctan\frac{1}{\phi^{10}} &= \arctan\frac{2\sqrt5}{275} \\
2\arctan\frac{1}{\phi^{11}} &= \arctan\frac{2}{199}
\end{align}$$
Here are the observations


*

*Odd powers of inverse $\phi$ in double arctan functions lead to arctan of well defined fractions

*Even powers of inverse $\phi$ in double arctan functions lead to arctan of fractions involving $\sqrt5$.



My curiosity is to know, is there any pattern in these interesting series?

I will be grateful to understand more, if anyone has come across such evaluations.
 [/INST] Here is the pattern:
Given $$2\arctan\frac1{\phi^n}=\arctan A_n$$
the following recursive reciprocals hold, for both odd and even $n$'s,
$$\frac1{A_{n+2}}+\frac1{A_{n-2}}=\frac3{A_{n}}$$
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