一般地设(n<=m)
设正方形个数为(x),长方形个数为(y)
[x=sum_{i=1}^{n}(n-i+1)(m-i+1)
]
[x=sum_{i=1}^{n}(nm-mi+m-ni+i^2-i+n-i+1)
]
[x=n(nm+m+n+1)sum_{i=1}^{n}(i^2-(m+n+2)i)
]
[x=n(nm+m+n+1)+frac{n(n+1)(2n+1)}{6}-(m+n+2)frac{n(1+n)}{2}
]
[x+y=dbinom{n+1}{2}dbinom{m+1}{2}
]