Number pattern prediction algorithm Since childhood we are all familiar with the task of predicting the next number in a sequence.
From something simple like, $2,4,6,...$ and $4,9,25,...$ to
something more complex like, $0,1,2^2,4^2+1,(4^2+2)^2,(4^2+2)^2+1,((4^2+2)^2+2)^2$ which would look like this if it was asked as a question $0,1,4,5,36,37,1444$.
There (probably) is an infinite number of different possible patterns.
My question is, does an algorithm exist that can predict any type of pattern? And considering that I think that the answer is no, my other question is, are there any algorithms that can predict any pattern to a certain level of complexity?
EDIT
@Thomas Andrews Thank you for pointing out the flaw with my question.
In light of that, I would like to change my question to this:
Is there an algorithm to determine a finite (or infinite) list of possible functions that match the given sequence?
 A: Pattern recognition is a useful skill in a mathematician, but it is not actually mathematically rigorous. There is no strictly correct answer, but, if you encountered a sequence in the wild, on some problem, the continuation would be merely a conjecture.
Consider: What is the maximum number of regions you get by drawing all the chords formed by $n$ points on a circle? That sequence starts:
$$1,2,4,8,16$$
You'd guess the next number is $32$, but it turns out, the next number is $31$.
That doesn't mean that recognizing patterns is not useful, but it is useful only in generating a conjecture about the sequence you are looking at, not an actually rigorous answer.
When mathematicians confuse conjecture and pattern-recognition, you end up making errors. I was reading Dickson's "History of the Theory of Numbers" a few years back, and was amused how often conjectures were stated as facts in ancient and medieval times. Did you know that for natural number $n$, there is exactly one perfect number with $n$ digits? It's not true, but was asserted as fact.
Modern mathematics is more careful. We might see that the first $n$ perfect numbers are all even, for some $n$, but we do not treat it as fact that all perfect numbers are even.
Every sequence of $n$ numbers can be written as a polynomial of degree $n-1$. Often, that isn't going to be the intuitive answer you wanted. For example, $4,9,25,\dots$ might be the squares of the primes, or it might be:
$$f(n)=\frac{11n^2-n+8}{2}$$
Which is it? It is neither, or it is both. The question does not have a single answer. Indeed, the next number could be made to be any integer.
