The coefficients are determined by substituting $p_n$ into the homogeneous part $a_n - 5a_{n-1} + 6a_{n-2} = 0$.

The coefficients $A, B, C$ must be calculated by substituting $p_n$ into the full non-homogeneous equation and equating coefficients of $n^k$.

The particular solution $p_n$ is a solution to the homogeneous equation $a_n - 5a_{n-1} + 6a_{n-2} = 0$.

The constants $A, B, C$ are determined solely by the initial conditions $a_0$ and $a_1$.