Find the recursive quadratic formula of the sequence: 1, 3, 7, 13, 21

The sequence

1, 3, 7, 13, 21, ...

has first-order differences

2, 4, 6, 8, ...

Let denote the original sequence, and the sequence of first-order differences. It's quite clear that

for . By definition of first-order differences, we have

for , or

By substitution, we have

and so on, down to

You should know that

and we're given , so

or

Alternatively, since we already know the sequence is supposed to be quadratic, we can look for coefficients such that

We have

and we can solve this system for the 3 unknowns to find .