$$\begin{cases} C_n=-C_{n-1}+C_{n-2}\\ C_0=1\\ C_1=0 \end{cases} $$
$ \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?
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.