> You can more easily generalize this solution to other arithmetic sequences.
The closed form as sums of exponents also easily generalizes. You can find the roots of an associated polynomial, and the those roots are the numbers you take powers of. This is explained on Wikipedia here: https://en.wikipedia.org/wiki/Recurrence_relation#Solving_ho....
As an aside, a lovely (freely available) book on using generating functions to calculate closed form solutions to such problems by the late Herbert Wilf: https://www.math.upenn.edu/~wilf/DownldGF.html
Nothing to add to this except to say that Generatingfunctionology is an awesome book. Although I haven't read it, another in the same space is A=B, also co-authored by Wilf (https://www.math.upenn.edu/~wilf/AeqB.html) (another co-author is Zeilberger, whose name anyone interested in computational discrete math will recognise).
The closed form as sums of exponents also easily generalizes. You can find the roots of an associated polynomial, and the those roots are the numbers you take powers of. This is explained on Wikipedia here: https://en.wikipedia.org/wiki/Recurrence_relation#Solving_ho....