6.6.32 Combinatorics

Problem: Construct a linear homogeneous difference equation (of the minimum possible order) with constant real coefficients, the partial solutions of which are the functions \[ (-3)^{s}, 5^{s}, s \cdot 5^{s}, 12^{s} \cos \frac{11 \pi(s+1)}{6} \text {. } \]