This week's book giveaway is in the OCAJP 8 forum. We're giving away four copies of OCA Java SE 8 Programmer I Study Guide and have Edward Finegan & Robert Liguori on-line! See this thread for details.
Just to expand a little on why that program works for a = (2^31) - 1. Now, since the largest possible positive int value is (2^31) - 1, you're right that you would get numeric overflow if you tried to calculate it in that order: the (2^31) will overflow before you have a chance to subtract the 1. However, if you unwind the calculation that is being done in line 12 of the program, what is actually being calculated is essentially (2^30) + (2^30 - 1). So the final step is an addition, and no overflow takes place.
[edit:] In looking back over your question, I realized that maybe you're confused about a different aspect of the problem. In the loop iteration when i = 31, it is true that the calculation for a[i+2] is going to be wrong due to overflow (as will have been a[i+1]). But that doesn't matter, because what is being printed in that iteration is a[i], which was calculated two iterations back and is correct.