Sequence of averages

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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_1 = 0 , a 2 = 512 a_2 = 512 and for n 3 n\geq 3 , a n = a n 2 + a n 1 2 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.

342 340 341 343 344

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