A342372 - OEIS

%I #34 Apr 07 2021 09:30:38

%S 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,

%T 1224,288,0,1,49,735,4165,8869,6321,931,133,1,64,1344,11648,43136,

%U 64512,33024,4096,0,1,81,2268,27972,160866,423306,469800

%N 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.

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

%C 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.

%C 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.

%H Walter Trump, <a href="/A342372/b342372.txt">Table of n, a(n) for n = 1..222</a>

%H Walter Trump, <a href="/A342372/a342372_2.pdf">Semi-queen problem</a>

%F T(n,0) = 1.

%F T(n,1) = n^2.

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

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

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

%F T(2n,2n) = 0.

%e   1;

%e   1,  1;

%e   1,  4,   0;

%e   1,  9,   9,   3;

%e   1, 16,  48,  32,  0;

%e   1, 25, 150, 250, 75, 15;

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

%K tabl,nonn

%O 1,5

%A _Walter Trump_, Mar 09 2021