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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