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> 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 ro
by spekcular 7y ago
> 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_homogeneous_linear_recurrence_relations_with_constant_coefficients https://en.wikipedia.org/wiki/Recurrence_relation#Solving_ho....
- phonebucket 7y agoAs 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 https://www.math.upenn.edu/~wilf/DownldGF.html One of the more enjoyable maths texts I've read.
- JadeNB 7y agoNothing 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 https://www.math.upenn.edu/~wilf/AeqB.html) (another co-author is Zeilberger, whose name anyone interested in computational discrete math will recognise).
- phonebucket 7y agoI've spent some time on A = B as well. Very pleased to find someone else interested in these things on here!