AI 中文总结
研究计算\(n\times n\)格点正方形网格中矩形数量\(F(n)\)的方法，通过有理阶梯和等步骤，用\(O(M(N)\log N)\)次系数环运算等资源精确计算，还进行实验评估其与\(O(N^{3/2})\)全值算法性能。
AI 中文摘要
设\(F(n)\)为顶点属于\(n\times n\)格点正方形网格的矩形数量（不一定与坐标轴平行）。我们通过\(O(M(N)\log N)\)次系数环运算和\(O(N\log N)\)个工作内存的环元素精确计算完整表格\(F(1),\ldots,F(N)\)，其中\(M(N)\)是\(N\)次多项式乘法的常规界限。该算法在系数提取前应用平方根覆盖，通过局部分母分治递归评估所得有理楔形和三角和。通过莫比乌斯反演逐系数恢复原始方向，随后进行五次前缀和。实验评估了具有认证中国剩余定理（CRT）恢复的模数论变换（NTT）实现与\(O(N^{3/2})\)全值算法的性能。
英文摘要
Let $F(n)$ be the number of rectangles, not necessarily axis-parallel, whose vertices belong to the $n\times n$ square grid of lattice points. We compute the complete table $F(1),\ldots,F(N)$ exactly in $O(M(N)\log N)$ coefficient-ring operations and $O(N\log N)$ ring elements of working memory, where $M(N)$ is a regular bound for multiplying degree-$N$ polynomials. The ring-level statement assumes that $6$ is invertible; over $\mathbb Z$ the only division is instead performed exactly in the elementary boundary term. With quasi-linear polynomial multiplication the arithmetic bound is $O(N\log^2 N)$. The algorithm applies a square-root cover before coefficient extraction and evaluates the resulting rational wedge and triangular sums by a local-denominator divide-and-conquer recursion. Primitive directions are recovered coefficientwise by Möbius inversion, followed by five prefix sums. A modular number-theoretic-transform (NTT) implementation with certified Chinese-remainder (CRT) recovery is evaluated experimentally against the $O(N^{3/2})$ all-values algorithm.
