0,3,8,15,24

Pattern Identification:

15

24

The differences (3, 5, 7, 9) form an arithmetic sequence with a common difference of 2, indicating the original sequence is quadratic.

Each term is 1 less than a perfect square:

General Formula:

For the (n)-th term, the sequence follows:

[

a_n = n^2

]

Verification:

Using the quadratic sequence method, solving for coefficients confirms the formula (a_n = n^2

Conclusion:

The sequence is generated by the rule (a_n = n^2

(boxed{a_n = n^2 - 1})