<s>[INST] Finding $C_n=-C_{n-1}+C_{n-2}$ $$\begin{cases}
C_n=-C_{n-1}+C_{n-2}\\
C_0=1\\
C_1=0
\end{cases} $$
$
\begin{pmatrix} 
C_n \\
C_{n-1} 
\end{pmatrix}
\begin{pmatrix} 
-1 && 1 \\
1 && 0
\end{pmatrix}^n
\begin{pmatrix} 
1\\
0 
\end{pmatrix}$
$\begin{vmatrix} 
\lambda+1 && -1 \\
-1 && \lambda
\end{vmatrix}=\lambda^2+\lambda-1$ So $\lambda_{1,2}=-\frac{1}{2}\pm \frac{ \sqrt{5}}{2}$
$$C_n=a(-\frac{1}{2}+ \frac{ \sqrt{5}}{2})^nC_{n-1}+b(\frac{1}{2}- \frac{ \sqrt{5}}{2})^nC_{n-2}$$
$$\begin{cases}
1=a+b\\
0=a(-\frac{1}{2}+ \frac{ \sqrt{5}}{2})+b(\frac{1}{2}- \frac{ \sqrt{5}}{2})
\end{cases}$$
$C_n=\frac{1}{2}(-\frac{1}{2}+\frac{\sqrt{5}}{2})^n+\frac{1}{2}(\frac{1}{2}- \frac{ \sqrt{5}}{2})^n$
Is it correct?
 [/INST] Apart from an offset (shift of n to n-1) this is https://oeis.org/A039834 in the Online Encyclopedia of Integer Sequences, which provides an equivalent formula.
</s>