Finding the rule
Input: 9,24,3,39,12
Output:7,12,8,16,30
8 - (x-3)/6 + (4/5 + 703/9990)(x-3)(x-9) - (705/10000)(x-3)(x-9)(x-12) + (2586/1000000)(x-3)(x-9)(x-12)(x-24)
This is from Newton's divided difference formula, because I don't see any simple pattern emerging.