You could write:
Now, subtract the former from the latter to get a recursion (difference equation).
I'm not sure I fully understand how to do a recursive definition for an integer sequence containing an exponent.
I can figure out that:
which equates to (mutiples of 4) x 12...but I'm not sure how to finish this to make it a recursive definition. The question I'm trying to answer has different values, my hope is to learn how to do this in general and not the specific question I'm required to answer.
Finding an explicit formula for when is defined by recursion is a meaningful problem. It allows calculating directly, without going through . The converse problem — given an explicit formula, find the recursive definition — does not make much sense, or, rather, it is trivial. If , then where , i.e., g does not use the second argument. In this particular case, the recursive definition is not even simpler than the explicit formula .
Edit 1: The converse problem would make sense if in the function g can only depend on but not n. This restricted form of recursion is sometimes called iteration. But in this case is not a function of , i.e., the sequence cannot be defined using iteration.
Edit 2: To be more precise, every sequence defined by (primitive) recursion can also be defined by iteration, but the definition requires an intermediate sequence: first one defined by iteration and then for some function h.