> For the complete documentation index, see [llms.txt](https://docs.1l.lol/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.1l.lol/functions/recurrence.md).

# 递推

### 什么是递推

递推是从**已知的初始条件**出发，按某种规律（递推式）一步步推出后续结果。相比递归，递推一般用循环实现，效率更高。

### 典型例子：斐波那契数列

`f(1)=1, f(2)=1, f(n)=f(n-1)+f(n-2)`

```cpp
long long f[100];
f[1] = f[2] = 1;
for (int i = 3; i <= n; i++)
    f[i] = f[i - 1] + f[i - 2];
cout << f[n];
```

### 典型例子：爬楼梯

每次走 1 或 2 级，走到第 n 级有多少种走法？`f(n) = f(n-1) + f(n-2)`。

### 递推 vs 递归

|    | 递推      | 递归      |
| -- | ------- | ------- |
| 实现 | 循环，从前往后 | 函数自调用   |
| 效率 | 高       | 可能有重复计算 |
| 思路 | 由小推大    | 由大化小    |

### 推导递推式的步骤

1. 确定状态含义（f\[i] 表示什么）；
2. 确定初始值（如 f\[1]、f\[2]）；
3. 找出 f\[i] 与前面项的关系；
4. 用循环从小到大计算。

> 递推也是动态规划的基础，很多 DP 问题本质上就是递推。


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