Make a difference table like this:
Notice that I have simpl extended the pattern in the table to produce the next number in the sequence, namely 35.
The 2nd column lists the difference between 2 successive numbers in the first column; this column is called the first difference. The 3rd column is the difference between 2 successive numbers in the 2nd column and is called the 2nd difference.
I cannot prove this for you now, but I can tell you that whenever you see a difference table like this with the 2nd difference column constant then the 1st difference will be an arithmetic sequence and the first column will be a quadratic sequence.
In this case the the difference column is given by 2n + 1 where n ≥ 1. The original sequence is given by n2 - 1
for n ≥ 1.