A307304 Number of inequivalent ways of placing 2 nonattacking rooks on n X n board up to rotations and reflections of the board.

0, 1, 4, 13, 31, 66, 123, 214, 346, 535, 790, 1131, 1569, 2128, 2821, 3676, 4708, 5949, 7416, 9145, 11155, 13486, 16159, 19218, 22686, 26611, 31018, 35959, 41461, 47580, 54345, 61816, 70024, 79033, 88876, 99621, 111303, 123994, 137731, 152590, 168610, 185871

Offset: 1

Author

Mo Li, Apr 19 2019

Keywords

• nonn
• easy

Examples

			For n = 4 the a(4) = 13 solutions are
{{1,0,0,0}}  {{1,0,0,0}}  {{1,0,0,0}}
{{0,1,0,0}}  {{0,0,1,0}}  {{0,0,0,1}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
—————————————————————————————————————
{{1,0,0,0}}  {{1,0,0,0}}  {{1,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
{{0,0,1,0}}  {{0,0,0,1}}  {{0,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,1}}
—————————————————————————————————————
{{0,1,0,0}}  {{0,1,0,0}}  {{0,1,0,0}}
{{1,0,0,0}}  {{0,0,1,0}}  {{0,0,0,1}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
—————————————————————————————————————
{{0,1,0,0}}  {{0,1,0,0}}  {{0,1,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
{{0,0,1,0}}  {{0,0,0,1}}  {{0,0,0,0}}
{{0,0,0,0}}  {{0,0,0,0}}  {{0,0,1,0}}
—————————————————————————————————————
{{0,0,0,0}}
{{0,1,0,0}}
{{0,0,1,0}}
{{0,0,0,0}}
		

Links

• Leisure Maths Entertainment Forum, 2 nonattacking rooks on n X n board, Chinese blog.

Crossrefs

Cf. A000903, A163102, A035287, A179058, A144084.

Programs

Mathematica

Table[
Piecewise[{{(n (n^3 - 2 n^2 + 6 n - 4))/16, Mod[n, 2] == 0},
{((n - 1) (n^3 - n^2 + 5 n - 1))/16, Mod[n, 2] == 1}}],{n, 20}]

Formula

a(n) = (1/16)*n*(n^3-2n^2+6n-4) if n is even;

a(n) = (1/16)*(n-1)*(n^3-n^2+5n-1) if n is odd.

G.f.: -x^2*(x^2+1)*(x^2+x+1)/((x+1)^2*(x-1)^5). - Alois P. Heinz, Apr 26 2019