Problem
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.
Constraints
1 ≤ n ≤ 10,000. Counts fit signed 64-bit integers.
Examples
Input:
n = 3
Expected output:
expected = [0, 6, 28]Input:
long long n = 3;
Expected output:
std::vector<long long> expected = {0, 6, 28};Input:
long n = 3;
Expected output:
long[] expected = new long[] {0, 6, 28};Explanation
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.
Input:
n = 1
Expected output:
expected = [0]Input:
long long n = 1;
Expected output:
std::vector<long long> expected = {0};Input:
long n = 1;
Expected output:
long[] expected = new long[] {0};Explanation
Two distinct squares are required, but the board has only one.
References
This challenge is adapted for CodeUtility. Follow the statement and constraints above.