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.
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
The next term is -2.
The number 0 and the next is 2.
4
312 211 110 3-1=2 2-1=1 1-1=0
The next whole number after 1 is 2. However, the decimal after 1 can be an infinite chain of 0's, with a 1 at the end.
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) ...
This series is of the function f(x) = x2+1, starting with x=0.The next number in the series is 26. The number after that is 37.
The next number is 4, followed by -2
The series appears to be alternating between adding and subtracting numbers. The pattern seems to be adding 1, then subtracting 2, then adding 3, and finally subtracting 4. Following this pattern, the next number in the series would be 1 (subtracting 4 from -1).
It is: 0 1 2 3 ... etc
4.
5.