$g\left(n\right)=\underset{0}{\overset{n^2+n+1}{\int }}{e}^{x/2-\left[x/2\right]}\left(\frac{x}{2}-\left[\frac{x}{2}\right]\right)d\left(x-\left[x\right]\right) ;n\in N$  then   $g\left(n\right)$([.] denotes greatest integer function)

$g\left(n\right)=\underset{0}{\overset{n^2+n+1}{\int }}{e}^{\{x/2\}}\left\{\frac{x}{2}\right\}d\left\{x\right\}=(n^2+n+1)\underset{0}{\overset{1}{\int }}{e}^{\{x/2\}}\left\{\frac{x}{2}\right\}dx$

$=(n^2+n+1)\underset{0}{\overset{1}{\int }}{e}^{x/2}\left(\frac{x}{2}\right)dx=n^2+n+1[4-2e^{1/2}]$ So, minimum value is  $12-6\sqrt{e}$