My 2 cents (ha ha): perhaps the approach is to show that the power of 2 dividing $2+\sum_{i=1}^{n-1}a_i^2$ is eventually less than the power of 2 in $n$, and that this somehow involves looking at $a_{2^n}$. I'll keep playing with it for a bit.

(inspired by [this][1] kind of argument that the harmonic numbers are not integers)


  [1]: http://mathheaven.blogspot.com/2008/05/nth-harmonic-number-cannot-be-integer.html