cp's OEIS Frontend

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

%I A307304 #44 Mar 02 2025 10:03:58
%S A307304 0,1,4,13,31,66,123,214,346,535,790,1131,1569,2128,2821,3676,4708,
%T A307304 5949,7416,9145,11155,13486,16159,19218,22686,26611,31018,35959,41461,
%U A307304 47580,54345,61816,70024,79033,88876,99621,111303,123994,137731,152590,168610,185871
%N A307304 Number of inequivalent ways of placing 2 nonattacking rooks on n X n board up to rotations and reflections of the board.
%H A307304 Leisure Maths Entertainment Forum, <a href="https://kuing.cjhb.site/thread-6019-1-1.html">2 nonattacking rooks on n X n board</a>, Chinese blog.
%F A307304 a(n) = (1/16)*n*(n^3-2n^2+6n-4) if n is even;
%F A307304 a(n) = (1/16)*(n-1)*(n^3-n^2+5n-1) if n is odd.
%F A307304 G.f.: -x^2*(x^2+1)*(x^2+x+1)/((x+1)^2*(x-1)^5). - _Alois P. Heinz_, Apr 26 2019
%e A307304 For n = 4 the a(4) = 13 solutions are
%e A307304 {{1,0,0,0}}  {{1,0,0,0}}  {{1,0,0,0}}
%e A307304 {{0,1,0,0}}  {{0,0,1,0}}  {{0,0,0,1}}
%e A307304 {{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
%e A307304 {{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
%e A307304 —————————————————————————————————————
%e A307304 {{1,0,0,0}}  {{1,0,0,0}}  {{1,0,0,0}}
%e A307304 {{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
%e A307304 {{0,0,1,0}}  {{0,0,0,1}}  {{0,0,0,0}}
%e A307304 {{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,1}}
%e A307304 —————————————————————————————————————
%e A307304 {{0,1,0,0}}  {{0,1,0,0}}  {{0,1,0,0}}
%e A307304 {{1,0,0,0}}  {{0,0,1,0}}  {{0,0,0,1}}
%e A307304 {{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
%e A307304 {{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
%e A307304 —————————————————————————————————————
%e A307304 {{0,1,0,0}}  {{0,1,0,0}}  {{0,1,0,0}}
%e A307304 {{0,0,0,0}}  {{0,0,0,0}}  {{0,0,0,0}}
%e A307304 {{0,0,1,0}}  {{0,0,0,1}}  {{0,0,0,0}}
%e A307304 {{0,0,0,0}}  {{0,0,0,0}}  {{0,0,1,0}}
%e A307304 —————————————————————————————————————
%e A307304 {{0,0,0,0}}
%e A307304 {{0,1,0,0}}
%e A307304 {{0,0,1,0}}
%e A307304 {{0,0,0,0}}
%t A307304 Table[
%t A307304 Piecewise[{{(n (n^3 - 2 n^2 + 6 n - 4))/16, Mod[n, 2] == 0},
%t A307304 {((n - 1) (n^3 - n^2 + 5 n - 1))/16, Mod[n, 2] == 1}}],{n, 20}]
%Y A307304 Cf. A000903, A163102, A035287, A179058, A144084.
%K A307304 nonn,easy
%O A307304 1,3
%A A307304 _Mo Li_, Apr 19 2019