GAIC Math 2024 Problem 148

Let $n$ be represented as the difference of squares of two nonzero natural numbers, then there are $F(n)$ ways to do so.

For example, $15=8^{2}-7^{2}=4^{2}-1^{2}$, so $F(15)=2$; whereas 2 cannot be represented, hence $F(2)=0$. Then, the calculation result of $F(1)+F(2)+F(3)+\cdots+F(100)$ is _____.