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In some simple cases it is even possible to both guess some results and prove them too experimentally. I have done this with some Hankel determinants. Suppose you want to compute the Hankel determinant $\det \left( {a_{i + j} } \right)_{i,j = 0}^{n - 1}$ of a sequence $(a_n )$ . If no Hankel determinant vanishes then you can compute polynomials $p_n (x)$ which are orthogonal with respect to the linear functional $F$ on the polynomials defined by $F(x^n ) = a_n .$ By Favard's theorem there exist numbers $s_n ,t_n$ such that $p_{n + 2} (x) = (x - s_{n + 1} )p_{n + 1} (x) - t_n p_n (x).$ In some cases it is easy to guess a formula for these numbers after computing some of them. Thus also the Hankel determinant can be computed. In order to show that this guess gives the correct answer we can do the following: Define $a(n,j) = a(n - 1,j - 1) + s_j a(n - 1,j) + t_j a(n - 1,j + 1)$ with $a(0,j) = \delta _{j,0} .$ By the underlying theory it suffices to show that $a(n,0) = a_n .$ In order to do this we again compute $a(n,j)$ for small values and try to guess a closed formula for them. In many cases we succeed. Then it suffices to verify that $a(n,j) = a(n - 1,j - 1) + s_j a(n - 1,j) + t_j a(n - 1,j + 1)$ holds for the conjectured formula.