I have a series of randomly generated numbers (3, 1,-3, 2, -1,6). I want to prove that the numbers are not random but have an increasing/decreasing pattern.
The source of the numbers isn't really important. I just want to prove that the numbers are truly random.
It is surprisingly hard to formally define random and hence hard to prove whether something is random or not. For example, are the digits of $\pi$ random? They pass pretty much any test or randomness yet they are predictable from one of the many ways of calculating $\pi$.
Finite sequences are particularly troublesome since it is easy to construct a formula that will generate any specified finite sequence.
One attractive way to say whether a sequence is random is whether the most compact way to specify it is to just list the values. In his sense, the digits of $\pi$ fails since there are quite simple formulae which generate them. Note that I am not claiming this as a standard definition just one that is sometimes appropriate.