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A232567 Number of non-equivalent binary n X n matrices with two nonadjacent 1's.

%I A232567 #36 Mar 14 2018 03:48:00
%S A232567 0,1,6,17,43,84,159,262,426,635,940,1311,1821,2422,3213,4124,5284,
%T A232567 6597,8226,10045,12255,14696,17611,20802,24558,28639,33384,38507,
%U A232567 44401,50730,57945,65656,74376,83657,94078,105129,117459,130492,144951,160190,177010
%N A232567 Number of non-equivalent binary n X n matrices with two nonadjacent 1's.
%C A232567 Also: Number of non-equivalent ways to place two non-attacking wazirs on an n X n board.
%C A232567 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 A232567 This sequence counts equivalence classes induced by the dihedral group D_4. If equivalent matrices are distinguished, the number of matrices is A172225(n).
%H A232567 Heinrich Ludwig, <a href="/A232567/b232567.txt">Table of n, a(n) for n = 1..1000</a>
%H A232567 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1)
%F A232567 a(n) = (n^4 + 2*n^2 - 4*n)/16 if n is even; a(n) = (n^4 + 4*n^2 - 8*n + 3)/16 if n is odd.
%F A232567 G.f.: x * (1 + x + x^2)*(1 + 3*x - x^2 + x^3) / ((1 + x)^3*(1 - x)^5). - _Bruno Berselli_, Nov 28 2013
%e A232567 There are a(3) = 6 non-equivalent 3 X 3 matrices with two nonadjacent 1's (and no other 1's):
%e A232567   [1 0 0]    [0 1 0]    [1 0 0]    [0 1 0]    [1 0 1]    [1 0 0]
%e A232567   |0 0 0|    |0 0 0|    |0 1 0|    |1 0 0|    |0 0 0|    |0 0 1|
%e A232567   [0 0 1]    [0 1 0]    [0 0 0]    [0 0 0]    [0 0 0]    [0 0 0]
%o A232567 (PARI) x='x+O('x^99); concat(0, Vec(x*(1+x+x^2)*(1+3*x-x^2+x^3)/((1+x)^3*(1-x)^5))) \\ _Altug Alkan_, Mar 14 2018
%Y A232567 Cf. A232568, A232569, A239576, A201511, A172225.
%K A232567 nonn,easy
%O A232567 1,3
%A A232567 _Heinrich Ludwig_, Nov 26 2013