Consider a sequence C = (C0, C1, C2, …). Define:
A0,j = Cj,
Ai,j = Ai,j+1 − Ai,j, and
Ri = Ai,o.
This might be pictured:
R = (R0, R1, R2, …) is another sequence. We might call C the sequence generated by R and R the generator of C.
It turns out that
The first of these formulae is particularly handy if the the sequence R is finite (or rather, is eventually zero):
Given some unknown sequence, this formula may be used to find empirically the power-series closed form, if it has one.
For example, consider the tetrahedral numbers C = (0, 1, 4, 10, 20, 35, …). We find that R = (0, 1, 2, 1).
This is not a proof of course, but it can be a useful experimental tool. It is also useful for manipulating sum sequences and difference sequences.