%I #20 Oct 10 2017 22:34:30

%S 0,3,62,683,4015,16989,56196,158271,391917,882683,1836106,3587103,

%T 6638267,11747613,19985680,32879339,52490521,81638211,124000342,

%U 184440963,269135111,386033453,545007772,758491143,1041639045,1413189339,1895630946,2516334551,3307717267

%N Number of non-equivalent (mod D_4) binary n X n matrices with 4 pairwise nonadjacent 1's.

%C Also number of non-equivalent ways to place 4 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 Without the restriction "non-equivalent (mod D_4)" numbers are given by A172227.

%H Heinrich Ludwig, <a href="/A239576/b239576.txt">Table of n, a(n) for n = 2..1000</a>

%H <a href="/index/Rec#order_14">Index entries for linear recurrences with constant coefficients</a>, signature (4,-1,-16,19,20,-45,0,45,-20,-19,16,1,-4,1)

%F a(n) = (n^8 -30*n^6 +24*n^5 +352*n^4 -576*n^3 -1280*n^2 +3360*n -1536 + IF(n==1 mod 2)*(14*n^4 -72*n^3 +226*n^2 -624*n +717))/192; n>=3.

%F G.f.: -x^2 - x^2*(1 - x + 51*x^2 + 454*x^3 + 1374*x^4 + 2527*x^5 + 1990*x^6 + 634*x^7 - 347*x^8 - 49*x^9 + 87*x^10 + 16*x^11 - 20*x^12 + 3*x^13) / ((-1+x)^9*(1+x)^5). - _Vaclav Kotesovec_, Mar 29 2014

%e There are a(3) = 3 non-equivalent binary 3 X 3 matrices with 4 pairwise nonadjacent 1s (x):

%e [0 1 0] [1 0 1] [1 0 1]

%e |1 0 1| |0 1 0| |0 0 0|

%e [0 1 0] [1 0 0] [1 0 1]

%t Flatten[{0,Table[(n^8-30*n^6+24*n^5+352*n^4-576*n^3-1280*n^2+3360*n-1536+If[EvenQ[n],0,(14*n^4-72*n^3+226*n^2-624*n+717)])/192,{n,3,20}]}] (* _Vaclav Kotesovec_ after _Heinrich Ludwig_, Mar 28 2014 *)

%t Drop[CoefficientList[Series[-x^2 - x^2*(1 - x + 51*x^2 + 454*x^3 + 1374*x^4 + 2527*x^5 + 1990*x^6 + 634*x^7 - 347*x^8 - 49*x^9 + 87*x^10 + 16*x^11 - 20*x^12 + 3*x^13) / ((-1+x)^9*(1+x)^5), {x, 0, 20}], x],2] (* _Vaclav Kotesovec_, Mar 29 2014 *)

%Y Cf. A232569, A232568, A232567, A172227.

%K nonn,easy

%O 2,2

%A _Heinrich Ludwig_, Mar 28 2014