$\sum_{-\infty}^{+\infty}\frac{\exp(-n^2)}{1-4n^2}$ in closed form? I have accrossed this sum which is defined as :$\sum_{-\infty}^{+\infty}\frac{\exp(-n^2)}{1-4n^2}$ , Wolfram alpha assume that series is converges and gives $\approx 0.75229789\cdots$ , really I have tried to present that value in closed form using Inverse symbolic calculator but i didn't succeeded , Now my  question is :  to give the value of the titled series in the closed form ? 
 A: I found general formula. It's not closed form solution only approximation for the sum:
$$\sum _{j=-\infty }^{\infty } \frac{\exp \left(-j^2\right)}{1-4 j^2}\approx \frac{\pi  \sum _{k=1}^n \left({\text{erfi}\left(\frac{1}{2}\right)}+(-1)^k
   \text{erfi}\left(\frac{1}{2}-k i \pi \right)+(-1)^k \text{erfi}\left(\frac{1}{2}+k i \pi \right)\right)}{2 \sqrt[4]{e}}$$
for $n=1$ it's only correct for 18 digits.
for $n=2$ it's only correct for 40 digits.
for $n=3$ it's only correct for 70 digits.
for $n=4$ it's only correct for 109 digits.
if $n>4$ then it's  better approximation and more correct digits.
Derivation of the formula
Borrowing integral $(d7)$ form user Dr. Wolfgang Hintze and integrating with Mathematica help:
$$\int \frac{\exp \left(-x^2\right) \left(1-\frac{1}{2} \pi  \left| \sin (x)\right| \right)}{2 \sqrt{\pi }} \, dx=\\\frac{4 \sqrt[4]{e} \text{erf}(x)-\pi  \left(\text{erfi}\left(\frac{1}{2}-i x\right)+\text{erfi}\left(\frac{1}{2}+i x\right)\right)+2 \pi 
   \left(\text{erfi}\left(\frac{1}{2}-i x\right)+\text{erfi}\left(\frac{1}{2}+i x\right)\right) \theta (\sin (x))}{16 \sqrt[4]{e}}+C$$
taking limit in jumps points:
$$\left(\underset{x\to \pi ^-}{\text{lim}}\text{int}-\underset{x\to 0^+}{\text{lim}}\text{int}\right)+\left(\underset{x\to \infty
   }{\text{lim}}\text{int}-\underset{x\to (2 \pi )^+}{\text{lim}}\text{int}\right)+\left(\underset{x\to (2 \pi )^-}{\text{lim}}\text{int}-\underset{x\to \pi
   ^+}{\text{lim}}\text{int}\right)+\left(\underset{x\to (-2 \pi )^-}{\text{lim}}\text{int}-\underset{x\to -\infty
   }{\text{lim}}\text{int}\right)+\left(\underset{x\to 0^-}{\text{lim}}\text{int}-\underset{x\to (-\pi )^+}{\text{lim}}\text{int}\right)+\left(\underset{x\to
   (-\pi )^-}{\text{lim}}\text{int}-\underset{x\to (-2 \pi )^+}{\text{lim}}\text{int}\right)=\\\frac{1}{2}-\frac{\pi 
   \left(\text{erfi}\left(\frac{1}{2}\right)-\text{erfi}\left(\frac{1}{2}-i \pi \right)-\text{erfi}\left(\frac{1}{2}+i \pi
   \right)+\text{erfi}\left(\frac{1}{2}-2 i \pi \right)+\text{erfi}\left(\frac{1}{2}+2 i \pi \right)\right)}{4 \sqrt[4]{e}}$$
based on this, it was possible to deduce the formula.
