Find the solution to the following linear, homogeneous recurrence with constant coefficients:
aₙ ​=−4aₙ₋₁​−4aₙ₋₂ −16aₙ₋₃​ for n≥3 with initial conditions a0​=5,a₁​​=12,a₂=−40. The solution is of the form:

an​=(α+iβ)(ir)ⁿ +(α−iβ)(−ir)ⁿ +γsⁿ

for suitable integer constants α,β,γ,r,s. Note that the variable r in this problem doesn't represent a characteristic value. Find these constants and enter their values: