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%I #33 Feb 10 2024 09:22:30
%S 0,6,40,210,681,1919,4443,9481,18206,33164,56570,92996,146175,223565,
%T 330981,479779,678508,943586,1287036,1731654,2293765,3004011,3883935,
%U 4973645,6300906,7917064,9857198,12185816,14946491,18218969,22056585,26556551,31783320
%N Number of non-equivalent binary n X n matrices with three pairwise nonadjacent 1's.
%C Also: Number of non-equivalent ways to place three non-attacking wazirs on an n X n board.
%C Two matrix elements are considered adjacent if the difference of their row indices is 1 and the column indices are equal, or vice versa (von Neumann neighborhood).
%C Counted for this sequence are equivalence classes induced by the dihedral group D_4. If equivalent matrices are being destinguished, the number of matrices is A172226(n).
%H Heinrich Ludwig, <a href="/A232568/b232568.txt">Table of n, a(n) for n = 2..1001</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).
%F a(n) = (n^6 - 15*n^4 + 20*n^3 + 50*n^2 - 116*n + 48)/48 if n is even; a(n) = (n^6 - 15*n^4 + 28*n^3 + 29*n^2 - 76*n - 15)/48 if n is odd.
%F G.f.: x^3*(x^9-4*x^8+x^7+12*x^6+9*x^5-70*x^4-77*x^3-84*x^2-22*x-6) / ((x-1)^7*(x+1)^4). - _Colin Barker_, Dec 06 2013
%F a(n) = (n^6 - 15n^4 + 28n^3 + 29n^2 - 76n - 15 - ((n+1) mod 2) * (8n^3 - 21n^2 + 40n - 63))/48. - _Wesley Ivan Hurt_, Dec 06 2013
%e There are a(3) = 6 non-equivalent 3 X 3 matrices with three pairwise nonadjacent 1's (and no other 1's):
%e [1 0 0] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [0 1 0]
%e |0 1 0| |0 0 0| |0 0 1| |0 0 0| |0 1 0| |1 0 1|
%e [0 0 1] [1 0 0] [0 1 0] [0 1 0] [0 0 0] [0 0 0]
%p A232568:=n->(n^6-15*n^4+28*n^3+29*n^2-76*n-15-((n+1) mod 2)*(8*n^3-21*n^2+40*n-63))/48; seq(A232568(n), n=2..50); # _Wesley Ivan Hurt_, Dec 06 2013
%t Table[(n^6-15n^4+28n^3+29n^2-76n-15-Mod[n+1,2](8n^3-21n^2+40n-63))/48, {n, 2, 50}] (* _Wesley Ivan Hurt_, Dec 06 2013 *)
%Y Cf. A232567, A239576, A232569, A172226.
%K nonn,easy
%O 2,2
%A _Heinrich Ludwig_, Nov 28 2013
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