What is the field studying the search and generation of computer programs?
What you are describing is a Kolmogorov complexity approach. The Kolmogorov complexity of an integer sequence (or a string) is the size of the minimal program computing it (in some fixed Turing-complete language). Here we're looking for the minimal program which generates $n+1$ numbers, the first $n$ of which are the given sequence.

Kolmogorov complexity is not computable - that is, one cannot compute the size of a minimal program computing a given sequence. The implementation you mention computes a restricted version of Kolmogorov complexity in which the programs are very simple (and always halt).

Another (non-computable) approach is the universal probability distribution in which a given integer sequence of complexity $x$ is given a probability in proportion to $2^{-x}$. By conditioning on the first $n$ numbers produced, we can construct a probability distribution on the next number. By restricting the set of programs, we can make this distribution computable (though perhaps not efficiently).