Is anybody aware if there exists good computer software which tries to find, in a brute force manner, patterns in given finite sequences of numbers. For example , if you would give the Fibonacci sequence to it, it will see that there is a polynomial $P = x_1 + x_2 - x_3$, such that for all 3 consecutive integers in the sequence $x_1, x_2, x_3$ we have $P(x_1, x_2, x_3) = 0$.
Of course I understand that this is kind of an open question, furthermore it is slightly ill-posed, because one could always interpolate a finite sequence with a polynomial. But that polynomial will not be very nice in general. So, I guess the question is, more precisely, if there is a computer program (like sage or mathematica) which on input a sequence starts generating a list of algebraic relations satisfied by the elements, where the algebraic relations are of growing complexity in a suitable sense.