What is the nth term of the sequence 6 0 -6 -12?

There are infinitely many polynomials of order 4 (or higher) that will give these as the first four numbers and any one of these could be "the" rule. Short of reading the mind of the person who posed the question, there is no way of determining which of the infinitely many solutions is the "correct" one.

The simplest rule is a linear polynomial U(n) = 6*(2 - n) for n = 1 , 2, 3, ...