Can the formula for a series be "reverse engineered" given some of its terms? Say I have a sequence that goes:
12, 16, 21, 27, 34, 42, 51 and so on in a manner that the increment between $a_1$ and $a_2$ is 4, between $a_2$ and $a_3$ is is 5 etc.
The formula which describes this series is $(n(n+7)/2)+12$.
My question is: would there be a way to find out the formula above given I have only the "description of its behavior" or a number of its terms, say $a_1$ to $a_7$?
Now, I have a hunch that this may not be possible, given meme formulae such as $f(x)=18111/2x^4-90555x^3+633885/2x^2-452773x+217331$, which produces the series 1, 3, 5, 7, 217341, meaning that even the most "normal looking" sequences can go wild, and also this website (oeis.org) which seems to archive lots of sequences.
Still, I'm guessing there might be a way that produces one of the possible formulae, even if probably no method can infer them all.
Is it possible?
BTW, I don't have any higher education on Mathematics, which means, in my country, that with what I was given at high school, I would be ready to start taking Calculus 1 in college. Nevertheless, I'm very interested and often watch niche videos on YouTube without much trouble keeping up.
 A: At least as a place-holder response: given any finite sequence of numbers (real, complex, whatever), there is a definite algorithmic procedure to determine a polynomial $P$ whose values $P(1), ... , P(n)$ are those given entities. Likewise, if we want a linear combination of exponentials (instead of a linear combination of $x^n$'s), we can do that as well, though the determination of coefficients is far less elementary than in the polynomial case.
One form of the computations in the polynomial case use the finite-difference operator $\Delta(f)(x)=f(x+1)=f(x)$, and the sequence of polynomials that have the property that $\Delta f_{n+1}=f_n$. Namely, $x(x+1)(x+2)\ldots(x+n)$ or similar.
To be clear, though, given a finite sequence, there is truly no uniquely-determined "next" element in that sequence, unless there are restrictions. Sure, as in school exams and so on, sometimes there are "obvious", but, even then, given $1,2,4,8,16$, one exponential continuation is that the next thing is $32$, but the polynomial interpolation gives $31$.
EDIT: yes, there is also a notion of (Kolmogorov) complexity of a means/description of/for computing the sequence elements. So we might ask for "the/a simplest" formula/algorithm that reproduces the given finite sequence. And we might ask for a sort of meta-algorithm that produces that algorithm, given the finite sequence. Not trivial issues, although some versions are answerable.
