[Solution] From the given, we can see that the differences between $y$ are $4, 6, 8, 10$, indicating that it is a second-order arithmetic sequence, meaning the functional relationship should be of the form $y=a x^{2}+b x+c$.
First, we can eliminate $(A),(B),(D)$. Upon verification, $(C)$ fits.
Therefore, the answer is $(C)$.
5・13. As shown in the table, the relationship between $x$ and $y$ is
(A) $y=4 x-1$.
(B) $y=x^{3}-x^{2}+x+2$.
(C) $y=x^{2}+x+1$.
(D) $y=\left(x^{2}+x+1\right)(x-1)$.
(E) None of the above.
(3rd American High School Mathematics Examination, 1952)