A342372 Triangle T(n,k) of number of ways of arranging q nonattacking semi-queens on an n X n toroidal board, where 0 <= k <= n.

1, 1, 1, 1, 4, 0, 1, 9, 9, 3, 1, 16, 48, 32, 0, 1, 25, 150, 250, 75, 15, 1, 36, 360, 1200, 1224, 288, 0, 1, 49, 735, 4165, 8869, 6321, 931, 133, 1, 64, 1344, 11648, 43136, 64512, 33024, 4096, 0, 1, 81, 2268, 27972, 160866, 423306, 469800

Offset: 1

Author

Walter Trump, Mar 09 2021

Keywords

• tabl
• nonn

Comments

T(0,0):=1 for combinatorial reasons.

A semi-queen can only move horizontal, vertical and parallel to the main diagonal of the board. Moves parallel to the secondary diagonal are not allowed.

Instead of a board on a torus, you can imagine that the semi-queens can leave a flat board on one side and re-enter the board on the other side.

Examples

			  1;
  1,  1;
  1,  4,   0;
  1,  9,   9,   3;
  1, 16,  48,  32,  0;
  1, 25, 150, 250, 75, 15;
		

Links

• Walter Trump, Table of n, a(n) for n = 1..222
• Walter Trump, Semi-queen problem

Crossrefs

Cf. A006717, A099152, A103220, A202654, A202655, A202656, A202657.

Formula

T(n,0) = 1.

T(n,1) = n^2.

T(n,2) = n^2*(n-1)*(n-2)/2.

T(n,3) = n^2*(n-1)*(n-2)*(n^2-6n+10)/6.

T(2n+1,2n+1) = A006717(n).

T(2n,2n) = 0.