How to find a difference polynomial?

[datenshinoai]

Homework Statement

Let {a_n} be the sequence 3, 4, 8, 17,..., where a_0 = 3 and a_n+1 = a_n + (n+1)^2, n greater than or equal to 0. Find a polynomial such that a_n = f(n)

Homework Equations

f(n) = summation from r=0 to infinity of C(n,r) delta^r a_0

The Attempt at a Solution

I just have no idea how to start it. I have: f(n) = 3C(n, 0) + 4C(n, 1) + 8C(n,2) + something?

 

[Office_Shredder]

Look at the differences
4-3=1
8-4=4
17-8=9

1,4,9,... is just the sequence for n²

So f(n+1)-f(n) = n². If the difference has degree k, the actual polynomial has degree k+1 (something you should try proving) so find a cubic polynomial with this characteristic