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Given any number it is easy to find a rule based on a polynomial of order 5 such that the first five numbers are as listed in the question and the next is the given number. There are also non-polynomial solutions.

The simplest polynomial of order 4 is

Un = (-11n4 + 138n3 - 601n2 + 1074n - 624)/24 for n = 1, 2, 3, ...

and, accordingly, the next number is -11.

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Q: What is the next number -1 2 1 3 4?
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