For a general second-order linear recurrence equation f_(n + 1) = x f_n + y f_(n - 1), define a multiplication rule on ordered pairs by (A, B)(C, D) = (A D + B C + x A C, B D + y A C). The inverse is then given by (A, B)^(-1) = (-A, x A + B)/(B^2 + x A B - y A^2), and we have the identity (f_1, y f_0)(1, 0)^n = (f_(n + 1), y f_n) (Beeler et al. 1972, Item 12).
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