$$ F(n) = \sum_{i=0}^{n} F\left(\left\lfloor\frac{i}{5}\right\rfloor\right) $$
I encountered this odd looking functional equation, while perusing the site yesterday. I'd be interested in seeing a formal/elegant solution method to find $F(n)$. My solution method is a bit ad-hoc.
Hint: To actually find the solution, start off by graphing. From there, I'd see if there is an formal/elegant proof method to justify the intuition.
Here's a challenge question, can your solution method be easily generalized to handle,$ F(n) = \sum_{i=0}^{n} F\left(\left\lfloor\frac{i}{k}\right\rfloor\right) $, where $k$ is now a positive number?
(I intend to post an answer, but I'll wait about $3$ or more days)