$\begin{cases}
a_n\equiv 0\pmod{2},n\equiv 0\pmod{2}\\
a_n\equiv 1\pmod{2},n\equiv 1\pmod{2}
\end{cases}$
$\begin{cases}
a_n\equiv n^{{0+1}^{0+2}}\equiv n\pmod{5},n\equiv 0\pmod{4}\\
a_n\equiv n^{{1+1}^{1+2}}\equiv n^8\equiv n^4\pmod{5},n\equiv 1\pmod{4}\\
a_n\equiv n^{{2+1}^{0+2}}\equiv n^9\equiv n\pmod{5},n\equiv 2\pmod{4}\\
a_n\equiv n^{{3+1}^{1+2}}\equiv n^{64}\equiv n^4\pmod{5},n\equiv 3\pmod{4}\\
\end{cases}$
$\begin{cases}
a_n\equiv 5(0)-4(n)\equiv 6n\pmod{10},n\equiv 0\pmod{2}\\
a_n\equiv 5(1)-4(n^4)\equiv 6n^4+5\pmod{10},n\equiv 1\pmod{2}\\
\end{cases}$
$a_n\equiv 0,1,2,1,4,5,6,1,8,1\pmod{10},n\equiv 0,1,2,3,4,5,6,7,8,9\pmod{10}$
$\displaystyle \sum_{n=1}^{2018}a_n\equiv 201\times 9+8\equiv 7\pmod{10}$ |