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A307304 Number of inequivalent ways of placing 2 nonattacking rooks on n X n board up to rotations and reflections of the board. 0
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 (list; graph; refs; listen; history; text; internal format)
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 A116411 A333047

Adjacent sequences:  A307301 A307302 A307303 * A307305 A307306 A307307

KEYWORD

nonn,easy

AUTHOR

Mo Li, Apr 19 2019

STATUS

approved

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Last modified September 28 10:48 EDT 2021. Contains 347714 sequences. (Running on oeis4.)