Any number 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 six numbers are as listed in the question followed by the chosen next number. 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.
In this particular example, the simplest solution, based on a polynomial or order 5, is
t(n) = (6n^5 - 125n^4 + 920n^3 - 2995n^2 + 4474n - 2280)/120 for n = 1, 2, 3, ...
and, accordingly, the next term is -12.
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The next term is -2.
16' then 25 and then 36 et.seq. The clue is 0,1,4,9,16,25,36, ... = 0^(2) , 1(2) , 2^(2), 3^(2) , 4^(2) , 5^(2) , 6^(2), ... n^(2) ...
It is: 0 1 2 3 ... etc
4
12110 0r 1210