Interpret each recurrence polynomial as the weighted sum of tilings by squares and dominoes: a domino beginning at i contributes the monomial factor a_i. The recurrence partitions tilings by their final tile. Reflection sends a domino beginning at i to one beginning at n-i, so it bijects every forward monomial with the corresponding reverse monomial. This establishes the polynomial identity symbolically; the frozen support table is a finite n=10 replay and does not prove the arbitrary-n identity for all real coefficient families.
RESULT_JSON: {"base_cases":[{"length":0,"value":"1"},{"length":1,"value":"1"}],"partition":"each_tiling_ends_in_a_square_or_domino","recurrence":"F_(k+1)=F_k+a_k*F_(k-1)","reflection":"i_maps_to_n_minus_i_and_is_an_involution"}
