I'm trying to find $\left\lfloor\sum_{k = 1}^{n}{\varphi^{3k}}\right\rfloor$ mod $m$. $\varphi = \frac{1 + \sqrt{5}}{2}$ and $\varphi^3 = 2 + \sqrt{5}$.
But honestly I'm not even sure where to start. I can see spending some time and finding a pattern for $\varphi^{3k}$ but I I need a way to compute the summation in case $n$ is large.
Any help is appreciated. Thank you.
P.S. My background is not in this. Sorry if this question is easy.