类欧几里得算法
对于求和式 $f(a,b,c,n)=\sum_{i=0}^n \lfloor \frac{ai+b}{c} \rfloor$
当 $a \geq c$ 或 $b \geq c$ 时,设 $a'=a \; mod \; c$,$b'=b \; mod \; c$,有
$$\begin{align*} f(a,b,c,n) = & \sum_{i=0}^n \; \lfloor \frac{ai+b}{c} \rfloor \\ = & \sum_{i=0}^n \; \lfloor \frac{a'i+b'}{c} \rfloor + \lfloor \frac{a}{c} \rfloor \times i + \lfloor \frac{b}{c} \rfloor \\ = & \; f(a',b',c,n) + \frac{n(n+1)}{2} \times \lfloor \frac{a}{c} \rfloor + (n+1) \times \lfloor \frac{b}{c} \rfloor \end{align*}$$
当 $a<c$ 且 $b<c$ 时,设 $m= \lfloor \frac{an+b}{c} \rfloor$,有
$$\begin{align*} f(a,b,c,n) = & \sum_{i=0}^n \; \lfloor \frac{ai+b}{c} \rfloor \\ = & \sum_{j=1}^m \sum_{i=0}^n \; [ \lfloor \frac{ai+b}{c} \rfloor \geq j ] \\ = & \sum_{j=0}^{m-1} \sum_{i=0}^n \; [ \lfloor \frac{ai+b}{c} \rfloor \geq j+1 ] \\ = & \sum_{j=0}^{m-1} \sum_{i=0}^n \; [ ai \geq jc+c-b ] \\ = & \sum_{j=0}^{m-1} \sum_{i=0}^n \; [ ai > jc+c-b-1 ] \\ = & \sum_{j=0}^{m-1} \sum_{i=0}^n \; [ i > \lfloor \frac{jc+c-b-1}{a} \rfloor ] \\ = & \sum_{j=0}^{m-1} n- \lfloor \frac{jc+c-b-1}{a} \rfloor \\ = & \; nm - \sum_{j=0}^{m-1} \; \lfloor \frac{jc+c-b-1}{a} \rfloor \\ = & \; nm-f(c,c-b-1,a,m-1) \end{align*}$$
这与欧几里得算法通过取模缩小范围递归的思想相似,时间复杂度为 $O(log \; a)$
ll f(ll a, ll b, ll c, ll n) { if (!a) return b / c * (n + 1); if (a < c && b < c) { ll m = (a * n + b) / c; if (!m) return 0; return n * m - f(c, c - b - 1, a, m - 1); } if (n & 1) return f(a % c, b % c, c, n) + (n + 1) / 2 * n * (a / c) + (n + 1) * (b / c); return f(a % c, b % c, c, n) + n / 2 * (n + 1) * (a / c) + (n + 1) * (b / c); }