Problem 53990. Classify product/digit-sum sequences
Cody Problem 53120 involved a sequence in which a term is computed by multiplying the previous two terms and adding the digits of the product. In that problem the first two terms of the sequence were 1 and 2. The next four terms were 2, 4, 8, and 5.
What happens if the first two terms are changed? It turns out that these product/digit-sum sequences can be sorted into five groups. For reasons that will likely be apparent to those who solved Cody Problem 53120, the sequence there (and others like it) is assigned the number 163. The other four types of sequence are assigned the numbers 1, 9, 26, and 62.
Write a function to classify the product/digit-sum sequences given the first two terms a and b. To encourage solvers to find the pattern, loops are banned, and to allow for large inputs, a and b are specified as strings.
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