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Suppose \(A\) is the generating function for the sequence \(3, 5, 9, 15, 23, 33, \ldots\text{.}\)

1. Find a generating function (in terms of \(A\)) for the sequence of differences between terms.

2. Write the sequence of differences between terms and find a generating function for it (without referencing \(A\)).

3. Use your answers to parts (a) and (b) to find the generating function for the original sequence.