What about the following sequence?
If we take quotients of entries in the last row instead of differences, we see that every quotient is 4. So f''(n) is a geometric, not an arithmetic sequence (as in our earlier examples).
What is the ``antiderivative'' of 
? Maybe we can figure this out
if we can determined the ``derivative'' of .
So the ``antiderivative'' of is 
.
Now we can find a formula for the sequence.
This method works in general if 
is geometric.