If the zeroth term was one of the terms you used to solve for the coefficients, you get the constant term of the polynomial for free and can immediately reduce the system to deg(p)+1 equations in deg(p)+1 unknowns as shown.
If the characteristic has a multiple root, this step is modified slightly. If r is a root of multiplicity m, use (c1rn + c2nrn + c3n2rn + . . .
- cmnm-1rn) instead of simply (c1rn). For example, the sequence starting 5, 0, -4, 16, 144, 640, 2240, . . . satisfies the recursive relationship an = 6an-1 - 12an-2 + 8an-3. The characteristic polynomial has a triple root of 2 and the closed form formula an = 52n - 7n2n + 2n2*2n.
