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Correct option is D

$$\textbf{Recurrence Relation} \\\text{Multiply both sides of the series expansion } f(z)(1 - z - z^2) = 1: \\\left( \sum_{n=0}^\infty a_n z^n \right)(1 - z - z^2) = 1. \\\text{Expand the product:} \\\sum_{n=0}^\infty a_n z^n - \sum_{n=0}^\infty a_n z^{n+1} - \sum_{n=0}^\infty a_n z^{n+2} = 1. \\\text{Reindex the second and third terms to align powers of } z^n: \\\text{In the second term: replace } n \to n-1, \text{ giving } \sum_{n=1}^\infty a_{n-1} z^n, \\\text{In the third term: replace } n \to n-2, \text{ giving } \sum_{n=2}^\infty a_{n-2} z^n. \\\text{The equation becomes:} \\a_0 + (a_1 - a_0)z + \sum_{n=2}^\infty (a_n - a_{n-1} - a_{n-2})z^n = 1. \\\text{For this to hold for all powers of } z, \text{ the coefficients must satisfy:} \\a_0 = 1, \\a_1 - a_0 = 0 \implies a_1 = a_0 = 1, \\\text{For } n \geq 2: \quad a_n = a_{n-1} + a_{n-2}.$$

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