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"

Answer

Polynomials in the standard form (as a sum of monomials) are actually the 'wrong' sort of 'encoding' to do interpolation. This post explains how to very efficiently interpolate any polynomial function, without ever computing the polynomial coefficients! In short, we can easily express any given polynomial sequence as a Newton series by computing successive forward differences of the given sequence, and not only can we use this to efficiently compute subsequent terms in the sequence, we can also quickly convert this representation back to standard polynomial form, in the same way the linked post demonstrates for the sum of the first $n$ perfect cubes.

For your example the iterated forward differences are easily computed by hand:

 1,  2,  3,  4,  5,  6,100,659,...
 1,  1,  1,  1,  1, 94,559,...
 0,  0,  0,  0, 93,465,...
 0,  0,  0, 93,372,...
 0,  0, 93,279,...

 0, 93,186,...
93, 93,...

Which also means that you can use this forward difference technique (which works for any polynomial sequence such as $1,4,9,16,25,\cdots$) to justify that you can extend any given sequence polynomially as a reasonable pattern.

That said, if too few terms of a sequence are given (which is often the case), it is actually easier to describe it as simply a list with no pattern, because the intended pattern would have a description that is longer than just listing the few terms given.