Another of $\frac{1^2}{1^2}+\frac{1-2^2+3^2-4^2}{1+2^2+3^2+4^2}+\cdots=\frac{\pi}{4}$ type expressable in cube? Gregory and Leibniz formula
(1)
$$-\sum_{m=1}^{\infty}\frac{(-1)^m}{2m-1}=\frac{\pi}{4}$$
We found another series equivalent to (1)
This is expressed in term of square numbers
$$-\sum_{m}^{\infty}\frac{\sum_{n=1}^{3m-2}(-1)^nn^2
}{\sum_{n=1}^{3m-2}(+1)^nn^2}=\frac{\pi}{4}$$
$$\frac{1^2}{1^2}+\frac{1-2^2+3^2-4^2}{1+2^2+3^2+4^2}+\frac{1-2^2+3^2-4^2+5^2-6^2+7^2}{1^2+2^2+3^2+4^2+5^2+6^2+7^2}+\cdots=\frac{\pi}{4}$$
Is there another series equivalent to (1) but expressable in term of cube numbers?

In term of cube numbers:
$$\frac{1^3}{1^3}+\frac{1^3-2^3}{1^3+2^3}+\frac{1^3-2^3+3^3}{1^3+2^3+3^3}+\frac{1^3-2^3+3^3-4^3}{1^3+2^3+3^3+4^3}+\cdots=3-\frac{\pi^2}{4}$$
