Simple sequence made complicated

A sequence \(a_{n}\) satisfies the following equation:

\(a_{n} = 4a_{n-1} - 5a_{n-2} + 2a_{n-3}\), for all positive integers \(n\) not less than \(3\)

If \(a_{0} = 1, a_{1} = 2, a_{2} = 3,\)

Find \(a_{1000}\)

This question will be lengthy if you substitute for \(a_{3}, a_{4}, a_{5}, ...\) directly. Try to find the general form of \(a_{n}\)


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