<s>[INST] How to make this difference equation continuous? $$ F_n=F_{n-1}-F_{n-2} $$
How can I convert this oscillating sequence into a continuous function? IE get it in terms of n.
 [/INST] This sequence has exactly 6 values:
($a$,$\ $ $b$, $\ $ $b-a$, $\ $ $-a$, $\ $ $-b$, $\ $ $a-b$) in cycle.
So, there is a reason to build trigonometric function with period of $6$:
$$
F(n) = A \sin \frac{\pi n}{3} + B \cos\frac{ \pi n}{3},
$$

If $F_0 = a$, $F_1 = b$, then
$$
\left\{
\begin{array}{c}
A\cdot 0 + B\cdot 1 = a;\\
A\cdot \dfrac{\sqrt{3}}{2} + B \cdot \dfrac{1}{2} = b; \\
A\cdot \dfrac{\sqrt{3}}{2} - B \cdot \dfrac{1}{2} = b-a; \\
\end{array}
\right.
$$ 
$B=a$, $A=\dfrac{2b-a}{\sqrt{3}}$.
So,
$$
F(n) = \dfrac{2b-a}{\sqrt{3}} \cdot \sin \frac{\pi n}{3} \;+ \; a \cdot \cos\frac{\pi n}{3},
$$
where $a=F_0$, $b=F_1$.
</s>