2850.51 – What's The Next Term?

These problems typically have many possible solutions. Who is to know what weird idea was in the mind of the creator of this sequence? Many formulas might have been used.

The answer we like best is simply zero. Why not? Could be, could it not, that the author of this sequence wanted it to be zero from the seventh term on?

In fact there is a simple formula at work here. Hint: Subtract one from all the terms. I bet you see it now. The "official answer" is 344. The terms are $n^3 + 1$.