Any more of this type? $\sum_{j=1}^{k}j\phi^j=\phi^{2k}$

Question

Curious equations of phi

$\phi={1+\sqrt5\over 2}$, golden ratio

Here are two examples,

$2\phi^2+1\phi^1=\phi^4$

$4\phi^4+3\phi^3+2\phi^2+1\phi^1=\phi^8$

Answer 1

\begin{array}{rcrcrcrcrcrcrcr} S &=& \phi &+& 2\phi^{2} &+& 3\phi^{3} &+& \ldots &+& k\phi^{k} & \\ S\phi &=& & & \phi^{2} &+& 2\phi^{3} &+& \ldots &+& (k-1)\phi^{k} & +& k\phi^{k+1} \\ S(1-\phi) &=& \phi &+& \phi^{2} &+& \phi^{3} &+& \ldots &+& \phi^{k} & - & k\phi^{k+1} \\ &=& \frac{1-\phi^{k+1}}{1-\phi} &-& k\phi^{k+1} \\ S &=& \frac{1-\phi^{k+1}}{(1-\phi)^{2}} &-& \frac{k\phi^{k+1}}{1-\phi} \end{array}

Note that the mentioned equality holds for $k=2,4$ only.