We know that $F_n = 4F_{n-3} + F_{n-6} = 11F_{n-5} + F_{n-10}$ and there are countless other relationships between Fibonacci numbers. However I was examining $F_n = 6F_a + F_b$ where $a,b > 0$.
It appears that the only solutions are $\mathbf{8}$ ($a=1,b=2$), $\mathbf{13}$ ($a=2,b=1$), and $\mathbf{21}$ ($a=b=3$). As I numerically explored higher numbers, it looked like the probability of finding a fourth solution diminished rapidly. I tried some transformations using the Pisano period for modulo 6 without success.
So: are there no other solutions? Can it be proved? Is there a more general upper limit to this sort of relationship, not just for 6?
