Symmetry Function

Geometry Level 5

"There exists a 2D figure with exactly \(m\) line(s) of symmetry and rotational symmetry of order \(n\)."

Suppose the above statement is called \(T(m, n)\). Let the function \(f : \mathbb{Z_{\geq 0}} \times \mathbb{Z_{> 0}} \rightarrow \mathbb{Z}\) be defined as:

\[f(m ,n) = \begin{cases} m + n & \text{if } T(m, n) \text{ is true} \\ 0 & \text{if } T(m, n) \text{ is false} \\ \end{cases} \]

If \(g : \mathbb{Z_{>0}} \rightarrow \mathbb{Z}\) is defined as \(g(x) = \displaystyle \sum _{m = 0} ^{x} \sum _{n = 1} ^{x} f(m, n) \),
find \(\displaystyle \lim_{x \to \infty} \frac{g(x)}{x^{2}}\)


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