【题解】P3704 [SDOI2017] 数字表格

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[SDOI2017] 数字表格 \(_\texttt{P3704}\)

定义 \(\xi(i)\) 表示斐波那契数列的第 \(i\) 项，\(f_i\) 表示 \(i\le n,j\le m,\gcd(i,j) = i\) 的方案数量，则 \(f_i=\sum_{i|d}\mu\left(\frac di\right)\left[\frac nd\right]\left[\frac md\right]=\sum _{k}\mu (k)\left[\frac n{ik}\right]\left[\frac m{ik}\right]\)。

\[\begin{align*} Ans & = \prod_{i,j} \xi(\gcd(i,j)) \\ &= \prod_i \xi(i)^{f_i}\\ &= \Large {\prod_i {\xi(i)}^{\normalsize{\displaystyle \sum _{k}\mu (k)\left[\frac n{ik}\right]\left[\frac m{ik}\right]}}}\\ &= \Large {\prod_i \prod _{k}{\xi(i)}^{\normalsize{\displaystyle \mu (k)\left[\frac n{ik}\right]\left[\frac m{ik}\right]}}}\\ &= \large {\prod_i \prod _{k}{\xi(i)}^{{\displaystyle \mu (k)}^{\small \displaystyle\left[\frac n{ik}\right]\left[\frac m{ik}\right]}}}\\ \end{align*} \]