PolyRing

If we define $X_n = x^n + x^{-n}$ , then $X_0=2$ , $X_1 = 2019$ and $X_{n+1} + X_{n-1} \; = \; (x + x^{-1})X_n \; = \; 2019X_n \hspace{2cm} n \ge 1$ It is clear that $X_n$ is therefore an integer for each $n \ge 0$ , and it is easy to calculate the remainder of $X_n$ modulo $1000000007$ using this recurrence relation. Since $C = X_{2019}$ we obtain the answer $\boxed{764123220}$ .