Word Count : 269

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Let (A(x) = \sum_{n=0}^{\infty} a_n x^n) be the generating function of the sequence ({a_n}). The recurrence is [ a_n = 6a_{n-1} - 5a_{n-2} + 1, \quad a_0 = 2, , a_1 = 5. ] Multiply both sides of the recurrence by (x^n) and sum over (n \ge 2): [ \sum_{n=2}^{\infty} a_n x^n = \sum_{n=2}^{\infty} 6 a_{n-1} x^n - \sum_{n=2}^{\infty} 5 a_{n-2} x^n + \sum_{n=2}^{\infty} x^n. ] Rewriting ________ _______ ___ ________ __________ _____ _______ ___ _____ ______.
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