Is there a sum formula for the following sequence?
It is a Gaussian with $N$ terms, $x_{0}$ and $\sigma$ are real numbers, with $0<x_{0}<N$
$$\sum_{x=0}^{N-1} e^{\frac{-1}{\sigma} \:(x-x_{0})^{2}}$$
I was looking for a formula like these ones, but if not possible some approximation is Ok.
I doubt that you can easily tame this sum.
For large $\sigma$, the $n$ Gaussians are much overlapping, giving a Gaussian-like shape. But for small $\sigma$, the Gaussians are apart. For intermediate values, you get a partial overlap and the function is irregularly oscillating.
The curves below are obtained by varying $x_0$, for different $\sigma$.
There could be an approach by finding a smooth, unimodal function that crosses the curve at the inflection points, and subtract it to get the oscillating component.
Then by fitting a curve to the positions of the zeroes, one could find a phase function, which should be quasi-linear. Then by taking the ratio with a sinusoid applied to that phase function, one should find a smooth amplitude envelope. The approximation would have the form
$$u(x_0)+a(x_0)\cos(\phi(x_0))$$ where $u,a,\phi$ themselves depend on $\sigma$.