The Fibonacci Addition Identity states that: ${F}_{n}={F}_{m}{F}_{n-m+1}+{F}_{m-1}{F}_{n-m}$. This was useful in showing that: ${F}_{i+k}={F}_{k-2}{F}_{i+1}+{F}_{k-1}{F}_{i+2}$. However, I would like to use this result to express the same for ${F}_{i}$ and ${F}_{i+3}$, where we can express ${F}_{i+k}$ in some linear combination of ${F}_{i}$ and ${F}_{i+3}$. Is there any way to do this? I haven't been able to make use of the typical Fibonacci substitutions to make any progress.