You, You and only you

Number Theory Level pending

We define a sequence \(u(n)\) for \(n=0,1,2 ,\ldots\) as follows: \(u(0) = 0, u(1)=1\), and for each \( n\geq1\), \(u(n+1) \) is the smallest positive integer such that \(u(n+1)>u(n)\) and \(\{u(0),u(1),u(2) ,\ldots, u(n+1)\}\) contains no 3 elements which are in an arithmetic progression.

What is \( u (2016) \)?

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