One more very nice explanation:

http://math.stackexchange.com/a/675137/309722

For a linear polynomial $p$, you'll always have $p(n+1) - p(n)$ the same. If you write down a table

1 2 3 4 5 p(1) p(2) p(3) p(4) p(5)

which in your case would be this:

0 1 2 3 5 4 9 20

and then write the differences $p(2) - p(1)$, $p(3) - p(2)$, etc in a row beneath, you'd get (again in your case)

0 1 2 3 5 4 9 20 -1 5 11

That new row is called the "first differences". For a linear function, the entries would all be the same. You can also write down second differences:

0 1 2 3 5 4 9 20 -1 5 11 6 6 For a quadratic function, these second differences will all be the same. Your values actually all the same, so your function can be expressed as a quadratic.