How to easily create a polynomial function that gives a desired output? I am looking for an easy way (formula or algorithm) to create a polynomial function that gives the arbitrary preset output for the first values of x. For instance, the desired output can be $y = 1, 2, 3, 4, 5, 6, 100$ for $x = 1, 2, 3, 4, 5, 6, 7$. I can create one of the many functions by hand as: 

$y = x + (x-1)(x-2)(x-3)(x-4)(x-5)(x-6)\cdot \frac{93}{720}.$

But for other desired outputs it is really painstaking. For example, I am working now with the desired output $y = 2, 10, 12, 16, 17, 18, 19, 200, 300$ when $x = 1, 2, 3, 4, 5, 6, 7, 8, 9$.
Notes:  
$x$ is always a natural positive number ($x > 0$). It doesn't matter what happens for other values of $x$ (greater than the ones given).
- I am looking for a easy way, as I don't know / I don't have / I have no experience with math related software.
- Also for a polynomial (or similar simple) function, as my mathematical background is quite limited, so I don't know any gamma, delta or other functions, series, etc.
- This quest is simply for recreation, to show friends how there are different solutions to the problems of the type: "Find the next term in this succession" 

As Gabriel Romon commented: Lagrange interpolation does the trick. 
The polynomials form a vector space (which works very similar to the 3D coordinate space) where the coefficients work as coordinates and the monomials including $x^0=1$ as orthogonal standard base. The Lagrange polynomials form another one. They are constructed from your given sampling points ($x$-values). The nice thing is that each Lagrange polynomial is zero in all but one sampling points. And in this sampling point it has the value $1$. So you can multiply each of the Lagrange polynomials with the function value at the sampling point where it has a nonzero value and sum over all these polynomials.
This looks as easy as 
$$
y = \sum_{i=1}^n y_i \cdot \ell_i(x).
$$
As you can see there is no $x_i$ in this formula any more. This makes it possible to precompute the Lagrange polynomials $\ell_i$ and solve all your problems together in a secound step, as long as all functions are defined using the same sampling points.
