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Any two numbers that you choose can be the next number. It is easy to find a rule based on a polynomial of order 6 such that the first five numbers are as listed in the question followed by the chosen next two 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.

One possible rule for the given numbers, based on a polynomial of order 4, is

t(n) = (5n^4 - 58n^3 + 235n^2 - 380n + 213)/3 for n = 1, 2, 3, ...

and, accordingly, the next two numbers in the sequence are 115 and 393.

But it would be easy to find a rule so that the next two were 4 and 6, for example.

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Q: What is the next two terms in the sequence 5 3 9 7 21?
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