{ "data": { "question": { "questionId": "3271", "questionFrontendId": "3015", "categoryTitle": "Algorithms", "boundTopicId": 2611695, "title": "Count the Number of Houses at a Certain Distance I", "titleSlug": "count-the-number-of-houses-at-a-certain-distance-i", "content": "

You are given three positive integers n, x, and y.

\n\n

In a city, there exist houses numbered 1 to n connected by n streets. There is a street connecting the house numbered i with the house numbered i + 1 for all 1 <= i <= n - 1 . An additional street connects the house numbered x with the house numbered y.

\n\n

For each k, such that 1 <= k <= n, you need to find the number of pairs of houses (house1, house2) such that the minimum number of streets that need to be traveled to reach house2 from house1 is k.

\n\n

Return a 1-indexed array result of length n where result[k] represents the total number of pairs of houses such that the minimum streets required to reach one house from the other is k.

\n\n

Note that x and y can be equal.

\n\n

 

\n

Example 1:

\n\"\"\n
\nInput: n = 3, x = 1, y = 3\nOutput: [6,0,0]\nExplanation: Let's look at each pair of houses:\n- For the pair (1, 2), we can go from house 1 to house 2 directly.\n- For the pair (2, 1), we can go from house 2 to house 1 directly.\n- For the pair (1, 3), we can go from house 1 to house 3 directly.\n- For the pair (3, 1), we can go from house 3 to house 1 directly.\n- For the pair (2, 3), we can go from house 2 to house 3 directly.\n- For the pair (3, 2), we can go from house 3 to house 2 directly.\n
\n\n

Example 2:

\n\"\"\n
\nInput: n = 5, x = 2, y = 4\nOutput: [10,8,2,0,0]\nExplanation: For each distance k the pairs are:\n- For k == 1, the pairs are (1, 2), (2, 1), (2, 3), (3, 2), (2, 4), (4, 2), (3, 4), (4, 3), (4, 5), and (5, 4).\n- For k == 2, the pairs are (1, 3), (3, 1), (1, 4), (4, 1), (2, 5), (5, 2), (3, 5), and (5, 3).\n- For k == 3, the pairs are (1, 5), and (5, 1).\n- For k == 4 and k == 5, there are no pairs.\n
\n\n

Example 3:

\n\"\"\n
\nInput: n = 4, x = 1, y = 1\nOutput: [6,4,2,0]\nExplanation: For each distance k the pairs are:\n- For k == 1, the pairs are (1, 2), (2, 1), (2, 3), (3, 2), (3, 4), and (4, 3).\n- For k == 2, the pairs are (1, 3), (3, 1), (2, 4), and (4, 2).\n- For k == 3, the pairs are (1, 4), and (4, 1).\n- For k == 4, there are no pairs.\n
\n\n

 

\n

Constraints:

\n\n\n", "translatedTitle": "按距离统计房屋对数目 I", "translatedContent": "

给你三个 正整数 n 、x 和 y 。

\n\n

在城市中，存在编号从 1 到 n 的房屋，由 n 条街道相连。对所有 1 <= i < n ，都存在一条街道连接编号为 i 的房屋与编号为 i + 1 的房屋。另存在一条街道连接编号为 x 的房屋与编号为 y 的房屋。

\n\n

对于每个 k（1 <= k <= n），你需要找出所有满足要求的 房屋对 [house1, house2] ，即从 house1 到 house2 需要经过的 最少 街道数为 k 。

\n\n

返回一个下标从 1 开始且长度为 n 的数组 result ，其中 result[k] 表示所有满足要求的房屋对的数量，即从一个房屋到另一个房屋需要经过的 最少 街道数为 k 。

\n\n

注意，x 与 y 可以 相等 。

\n\n

 

\n\n

示例 1：

\n\"\"\n
\n输入：n = 3, x = 1, y = 3\n输出：[6,0,0]\n解释：让我们检视每个房屋对\n- 对于房屋对 (1, 2)，可以直接从房屋 1 到房屋 2。\n- 对于房屋对 (2, 1)，可以直接从房屋 2 到房屋 1。\n- 对于房屋对 (1, 3)，可以直接从房屋 1 到房屋 3。\n- 对于房屋对 (3, 1)，可以直接从房屋 3 到房屋 1。\n- 对于房屋对 (2, 3)，可以直接从房屋 2 到房屋 3。\n- 对于房屋对 (3, 2)，可以直接从房屋 3 到房屋 2。\n
\n\n

示例 2：

\n\"\"\n
\n输入：n = 5, x = 2, y = 4\n输出：[10,8,2,0,0]\n解释：对于每个距离 k ，满足要求的房屋对如下：\n- 对于 k == 1，满足要求的房屋对有 (1, 2), (2, 1), (2, 3), (3, 2), (2, 4), (4, 2), (3, 4), (4, 3), (4, 5), 以及 (5, 4)。\n- 对于 k == 2，满足要求的房屋对有 (1, 3), (3, 1), (1, 4), (4, 1), (2, 5), (5, 2), (3, 5), 以及 (5, 3)。\n- 对于 k == 3，满足要求的房屋对有 (1, 5)，以及 (5, 1) 。\n- 对于 k == 4 和 k == 5，不存在满足要求的房屋对。\n
\n\n

示例 3：

\n\"\"\n
\n输入：n = 4, x = 1, y = 1\n输出：[6,4,2,0]\n解释：对于每个距离 k ，满足要求的房屋对如下：\n- 对于 k == 1，满足要求的房屋对有 (1, 2), (2, 1), (2, 3), (3, 2), (3, 4), 以及 (4, 3)。\n- 对于 k == 2，满足要求的房屋对有 (1, 3), (3, 1), (2, 4), 以及 (4, 2)。\n- 对于 k == 3，满足要求的房屋对有 (1, 4), 以及 (4, 1)。\n- 对于 k == 4，不存在满足要求的房屋对。\n
\n\n

 

\n\n

提示：

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