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

1,3

Links

Table of n, a(n) for n=1..42.

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

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

Example

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}}

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}]

Crossrefs

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

Sequence in context: A098536 A216563 A011937 * A097122 A378850 A116411

Adjacent sequences: A307301 A307302 A307303 * A307305 A307306 A307307

Keyword

nonn,easy

Author

Mo Li, Apr 19 2019

Status

approved