You are given a collection of integers (a finite set $S\subset\mathbb{Z}$). I'm interested in determining if there are some integers $a,b,c$ with $a\ge1$ such that for all $n\in S$ there is some $k_n\in\mathbb{Z}$ with $n = ak_n^2+bk_n+c,$ or proving that there are no such $(a,b,c).$ Any ideas/references/etc.?
If I knew the leading coefficient $a$ I could take a large member $\ell\in S$ and approximate its index as $$ k_\ell \approx \sqrt{\ell/a} $$ and doing this with several other large members would allow solving and guessing at $b$ and $c$. But since it's unconstrained there are too many free variables for such an approach to work. But it seems like the condition of fitting a single integer quadratic should be constraining enough that the problem should be solvable, I just need a better approach, one leaning more heavily on the Diophantine structure.
Hint:
If you subtract two of the numbers, the result is of the form
$$\Delta_1=a(k_n^2-k_m^2)+b(k_n-k_m)=(a(k_n+k_m)+b)(k_n-k_m).$$
Now if you try the divisors of $\Delta_1$, including $1$, you get the numbers $a(k_n+k_m)+b$ and your new task is to identify $a,b$.
Computing the differences between these new numbers
$$\Delta_2=a(l_i-l_j),$$ $a$ must be a divisor. Unfortunately, there are many combinations to be tried. Look for the differences with the less divisors.