While trying to answer this MSE question, I found that arctangents of many odd powers of the golden ratio $\varphi=\frac{1+\sqrt5}2$ are expressible as rational linear combinations of arctangents of positive integers:
$$\begin{align}
\arctan\varphi&=2\arctan1-\frac12\,\arctan2\\
\arctan\varphi^3&=\arctan1+\frac12\,\arctan2\\
\arctan\varphi^5&=4\arctan1-\frac32\,\arctan2\\
\arctan\varphi^7&=3\arctan1+\frac12\,\arctan2-\arctan5\\
\arctan\varphi^9&=\arctan1-\frac12\,\arctan2+\arctan4\\
\arctan\varphi^{11}&=5\arctan1+\frac12\,\arctan2-\arctan5-\arctan34\\
\arctan\varphi^{13}&=3\arctan1-\frac12\,\arctan2+\arctan4-\arctan89\\
\arctan\varphi^{15}&=-2\arctan1+\frac32\,\arctan2+\arctan11
\end{align}$$
I was not able to find such a representation for $\arctan\varphi^{17}$ though.
Question 1. Can we prove that it does not exist?
I also could not find such a representation for any positive even power.
Question 2. Can we prove that it does not exist for any positive even power?
Question 3. Is there a simple way to determine if such a representation exists for a given power?
Best Answer
Q1, Q2: Such a representation exists for all odd powers of $\varphi$ as we can show by induction. Using the arctangent identity, let us first write:
$\arctan \varphi^{2n+1} = \arctan\frac{\varphi^{2n+1}+v}{1-\varphi^{2n+1}v} - \arctan v$
Next, we set the argument $\frac{\varphi^{2n+1}+v}{1-\varphi^{2n+1}v}$ equal to $\varphi^{2n-1}$ in order to set up an induction and solve for $v$. After moving terms around and simplifying via identities involving $\varphi$, we end up with $v = \frac{\varphi^{2n-1}-\varphi^{2n+1}}{1+\varphi^{4n}} = -\frac{1}{L_{2n}}$ where $L_n$ is the $n^{th}$ Lucas number. As $\arctan\left(-\frac{1}{x}\right) = -(2\arctan 1-\arctan x)$, this proves the claim for odd powers of $\varphi$.
Q3: For even powers if we set up the analogous equation:
$\arctan \varphi^{2n} = \arctan\frac{\varphi^{2n}+v}{1-\varphi^{2n}v} - \arctan v$
However in this case, setting $\frac{\varphi^{2n}+v}{1-\varphi^{2n}v} = \varphi^{2n-2}$ and solving for $v$ gives $v=\frac{-1}{F_{2n-1}\sqrt{5}}$. This implies there cannot be a rational expression for $\arctan\varphi^{2n}$ in terms of $\arctan\varphi^{2n-2}$ and arctangents of integers for, if there were such an expression, the arctangent identity could then be applied to these terms to express $\frac{-1}{F_{2n-1}\sqrt{5}}$ as a rational number. Similar problems with $\sqrt{5}$ appearing in the expression for $v$ occur if we try setting the argument $\frac{\varphi^{2n}+v}{1-\varphi^{2n}v} = \varphi^k$ for other powers $k$ such as $k=2n-1$, so if any $\arctan \varphi^{2n}$ happen to be rational combinations of arctangents of integers, such expressions ought to be unrelated to one another unlike the situation for odd powers.
As a side note, this difference between the even and odd powers reminds me of the situation of values of the $\zeta$ function at integers where the $\zeta(2n)$ have simple closed form expressions while it is unknown if any $\zeta(2n+1)$ have a closed form expression. Perhaps someone with more familiarity of $\zeta$ values can comment on whether we might expect such similar behavior here.