Sequence of averages

Level pending

A sequence has first term 0 and second term 512, and after that each term is the arithmetic mean of the two previous terms, that is \(a_1 = 0\), \(a_2 = 512\) and for \(n\geq 3\), \(a_n = \frac{a_{n-2} + a_{n-1}}{2}\). What is the last term of the sequence that is an integer?

For a challenge, try to solve the problem without calculating all the integer terms.

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