What is the value of $f(0)+f(8)$? Suppose $f$ is a polynomial of degree $7$ which satisfies $f(1) =2$, $f(2)=5$, $f(3)=10$, $f(4)=17$, $f(5)=26$, $f(6)=37$ and $f(7)=50$. What is the value of $f(0)+f(8)$?

Note that the differences between the given values form an arithmetic sequence:
$$ +3,\ +5,\ +7,\ +9,\ +11,\ +13\,.$$
So that, we have an easy solution $f$, having $f(0)=1$ and $f(8)=65$. 
(Anyway, $f(x)=x^2+1$.)
Note that $7$ values determine uniquely a polynomial of degree $\le \bf 6$.
So, any other polynomial that satisfies the conditions is of the form
$$f(x)= x^2-1\ + \ g(x)\cdot(x-1)(x-2)\dots(x-7)\,.$$
