104. Define the dependence of variable $y$ on variable $x$ by a formula based on the table:

| $x$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $y$ | -1 | -3 | 1 | 17 | 51 | 109 | 197 | 321 | 487 |  |
Solution. Let's determine the differences of the given sequence:

$$
\begin{gathered}
-1, \quad-3, \quad 1, \quad 17, \quad 51, \quad 109,197,421,487, \quad \cdots \\
-2, \quad 4,16, \quad 34, \quad 58, \quad 88,124, \quad 166, \quad \cdots \\
6, \quad 12, \quad 18, \quad 24, \quad 30, \quad 36, \quad 42, \ldots
\end{gathered}
$$

We will not continue the calculations further, as the numbers in the third row form an arithmetic progression, which is a sequence of the 1st order. From this, it follows (see Theorem 2) that the second row is a sequence of the 2nd order, and the first row (the given sequence) is a sequence of the 3rd order.

Thus, the dependence of the value $y$ on the value $x$ can be expressed by a polynomial of the third degree:

$$
y=a x^{3}+b x^{2}+c x+d
$$

Now it remains to determine the coefficients $a, b, c$, and $d$ so that this polynomial takes the given values when $x=1,2,3, \ldots, 9$. Since it is necessary to determine four coefficients, we will choose four values of $x$, for example, $x=1,2,3,4$. Substituting these values of $x$ and the corresponding values of $y$ from the table into equation (5), we obtain a system of equations in terms of the variables $a, b, c$, and $d$:

$$
\left\{\begin{aligned}
a+b+c+d & =-1 \\
8 a+4 b+2 c+d & =-3 \\
27 a+9 b+3 c+d & =1 \\
64 a+16 b+4 c+d & =17
\end{aligned}\right.
$$

Solving this system (by the method of elimination of unknowns), we obtained: $a=1, b=-3, c=0, d=1$.

Therefore (see (5)), $y=x^{3}-3 x^{2}+1$.