問題
For each board size k from 1 through n, count unordered pairs of distinct squares on a k by k board where two knights cannot attack each other. A knight moves two squares in one axis and one in the other.
. x . x .
x . . . x
. . K . .
x . . . x
. x . x .K marks a knight and x its possible destinations, when those squares lie on the board. Swapping the two knights does not create another placement.
Implement two_knights(n).
Input
n: The largest board size to consider.
Output
Return n counts in order, for board sizes 1 through n.
制約
1 ≤ n ≤ 10,000. Counts fit signed 64-bit integers.
例
入力：
n = 3
期待される出力：
expected = [0, 6, 28]入力：
long long n = 3;
期待される出力：
std::vector<long long> expected = {0, 6, 28};入力：
long n = 3;
期待される出力：
long[] expected = new long[] {0, 6, 28};解説
A 1 × 1 board cannot hold two knights. A 2 × 2 board has six pairs and no attacks. A 3 × 3 board has 36 pairs, of which eight attack, leaving 28.
入力：
n = 1
期待される出力：
expected = [0]入力：
long long n = 1;
期待される出力：
std::vector<long long> expected = {0};入力：
long n = 1;
期待される出力：
long[] expected = new long[] {0};解説
Two distinct squares are required, but the board has only one.
解き方のヒント
まずは自分で考えてから、ヒントを 1 つずつ開いてみましょう。
ヒント 1
Start with all unordered pairs of squares on each board, then subtract the pairs where knights attack.
ヒント 2
Only a small, fixed number of knight attack placements fit inside any 2 by 3 or 3 by 2 rectangle.
ヒント 3
Edge boards like 1 by 1 and 2 by 2 need to naturally produce zero attacking pairs.
目安の計算量
O(n) time, O(1) extra space
参考資料
この課題はCodeUtility用に調整されています。上記の説明と制約に従ってください。