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Given any number it is easy to find a rule based on a polynomial of order 7 such that the first seven numbers are as listed in the question and the eighth is the given number. That is, it is easy to find a rule so that any number can be the "next" in the sequence.
However, the simplest rule here is the following polynomial of order 3:
Un = (n3 - 6n2 + 71n - 6)/6 for n = 1, 2, 3, ...
Accordingly, U8 = 115.
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