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\displaystyle I(n)= \[ \left\{\begin{array}{ll} 2 \lfloor{\frac{n}{\pi }} \rfloor +1-\cos n & \text{if }\lfloor \frac{n}{\pi } \rfloor \text{ is even} \\ 2\lfloor \frac{n}{\pi } \rfloor+1-\cos n & \text{if } \lfloor \frac{n}{\pi } \rfloor \text{ is odd} \\\end{array} \right. \]
\displaystyle i(m)= \[ \left\{\begin{array}{ll} 2 \lfloor{-\frac{m}{\pi }} \rfloor +1-\cos m & \text{if }\lfloor \frac{m}{\pi }+1 \rfloor \text{ is even} \\ 2\lfloor -\frac{m}{\pi } \rfloor+1-\cos n & \text{if } \lfloor \frac{m}{\pi }+1 \rfloor \text{ is odd} \\\end{array} \right. \]
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