I've heard (and believed even without proof) that given any finite sequence there is more than one formula for which the same first inputs give the same first outputs.
f(1)=1 f(2)=3 f(3)=6
One possible function is f(x) = x(x+1)/2.
So, I tried to find another polynomial function for which f(1)=1, f(2)=3 and f(3)=6, but for which f(4) is not 10, but got no success. So I ask: Can it be done? (please provide an example :))
I find this question interesting because often we can make a conjecture by finite induction which is wrong (like the proposed "prime-producing polynomials" which always start to fail after some time :P)