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Any number (or numbers) that you choose can be next. 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 followed by the chosen next numbers. There are also non-polynomial solutions. Short of reading the mind of the person who posed the question, there is no way of determining which of the infinitely many solutions is the "correct" one.

The simplest rule, based on a polynomial of order 4 is

T(n) = (-7n^4 + 80n^3 - 314n^2 + 499n - 246)/6 for n = 1, 2, 3, ...

Accordingly, the next three numbers are -58, -251 and -677.

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Q: What is the next numbers to the pattern 2 4 3 9 4?
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