We determine the chromatic polynomial of G_n= Fₙ □ P₃, where Fₙ is the friendship graph with n triangles and P₃ is the three-vertex path. Conditioning on proper colorings of the central path yields two inequivalent states, depending on whether the endpoint colors are distinct or equal. In either state, the remaining graph decomposes into n independent isomorphic six-vertex blocks. Exact enumeration of one block produces two degree-six local factors that yield an explicit two-term closed form and constitute the eigenvalues of a two-dimensional transfer operator. Cayley–Hamilton then gives a second-order linear recurrence for P_n(k). For real k > 3, λ₁(k) is strictly dominant, while λ₀(3) = λ₁(3) = 2, determining asymptotic growth and per-vertex chromatic entropy. Sturm-sequence analysis shows that λ₀has no real zeros and λ₁has exactly two. Every non-fixed accumulation point of chromatic roots lies on the equimodular set |λ₀(k)| = |λ₁(k)|.