integer sequences

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.

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One could always write something that will query the OEIS for you... as for Mathematica, it has a function called FindSequenceFunction[]. – J. M. Jul 13 '12 at 15:24
Wolfram Alpha can sort of do this, for example wolframalpha.com/input/… – Cocopuffs Jul 13 '12 at 15:24
Your explicit example $P(x_1,x_2,x_3)=0$ at W|A... – draks ... Jul 13 '12 at 16:33
Douglas Hofstadter has written long ago about Seek Whence, a program that tried to do that. I believe it was in his Metamagical Themas book. – Ross Millikan Oct 10 '12 at 19:27
+1 for the humor in saying that this problem is "slightly" ill-posed :). And I second the reference to Hofstadter... his treatment shows why this is as rich a question as any in artificial intelligence, and not likely to be cracked by brute force. – mjqxxxx Oct 10 '12 at 19:34