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Numbers f(1989), f(1990), f(1991) are divisible by 13

Source: IMO ShortList 1990, Problem 7 (GRE 2)

August 15, 2008
algebrapolynomialarithmetic sequencenumber theoryfunctional equationDivisibilityIMO Shortlist

Problem Statement

Let f(0) \equal{} f(1) \equal{} 0 and f(n\plus{}2) \equal{} 4^{n\plus{}2} \cdot f(n\plus{}1) \minus{} 16^{n\plus{}1} \cdot f(n) \plus{} n \cdot 2^{n^2},   n \equal{} 0, 1, 2, \ldots Show that the numbers f(1989),f(1990),f(1991) f(1989), f(1990), f(1991) are divisible by 13. 13.
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