I was looking at method of differences to solve a next in sequence problem. The method is applicable to sequences that can be expressed as a polynomial.
I started thinking about sequences whose sign of difference changes for every consecutive integer change.
Something like 5 4 6 5 7
Here the difference is -1 then +2. A series like this requires some kind of periodic function but then it will no longer be a polynomial.
If the series cannot be expressed as a polynomial, what kind of method can be used to find the next in sequence for sequences having difference like -1 +2 +3 -4 | -1 +2 +3 -4 | -1 .. or -2 +4 | -2 +4 ..
If $p$ is a polynomial such that $p(n+1)-p(n), n=1,2,...$ is alternately positive and negative the $p'$ has a zero in $(n,n+2)$ for every $n$. Thus $p'$ is a polynomial with infinitely many zeros which means $p'=0$ and $p$ is a constant.