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A207724 Number of n X 3 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.

%I A207724 #23 May 15 2023 08:44:38
%S A207724 6,36,78,189,490,1113,2449,5474,12036,26100,56595,122472,264061,
%T A207724 568695,1224190,2633000,5660226,12165489,26141850,56165805,120662377,
%U A207724 259206930,556801480,1196027864,2569059663,5518244160,11852866905,25459111647
%N A207724 Number of n X 3 0..1 arrays avoiding 0 0 0 and 0 1 0 horizontally and 0 1 1 and 1 0 1 vertically.
%C A207724 Column 3 of A207729.
%H A207724 R. H. Hardin, <a href="/A207724/b207724.txt">Table of n, a(n) for n = 1..210</a>
%H A207724 Robert Israel, <a href="/A207724/a207724.pdf">Maple-assisted proof of formula</a>
%H A207724 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,3,-6,0,0,3,1,0,-1).
%F A207724 Empirical: a(n) = 3*a(n-1) - 2*a(n-2) + 3*a(n-3) - 6*a(n-4) + 3*a(n-7) + a(n-8) - a(n-10).
%F A207724 Formula confirmed by _Robert Israel_, Mar 05 2018 (see link).
%F A207724 G.f.: x*(6 + 18*x - 18*x^2 + 9*x^3 + 7*x^4 + 3*x^5 - 9*x^6 - x^7 - x^8 + x^9) / ((1 - x)*(1 + x^2 - x^3)*(1 - x - x^3)*(1 - x - 2*x^2 - x^3)). - _Colin Barker_, Mar 05 2018
%e A207724 Some solutions for n=4:
%e A207724   1 1 0   1 1 1   1 0 0   1 0 0   0 1 1   0 1 1   0 0 1
%e A207724   1 0 1   1 1 1   1 0 0   1 1 1   0 1 1   1 1 1   1 1 1
%e A207724   1 0 0   1 1 0   1 0 0   1 0 0   1 1 1   0 1 1   0 0 1
%e A207724   1 1 0   1 0 0   1 1 1   1 0 0   0 0 1   0 1 1   0 0 1
%p A207724 Configs:= select(A -> A[1..3] <> [0,0,0] and A[4..6] <> [0,0,0] and A[1..3] <> [0,1,0] and A[4..6] <> [0,1,0],
%p A207724 [seq(convert(x,base,2)[1..6],x=2^6..2^7-1)]):
%p A207724 compat:= proc(i,j) local k,col;
%p A207724    if Configs[i][4..6] <> Configs[j][1..3] then return 0 fi;
%p A207724    for k from 1 to 3 do
%p A207724       col:= [Configs[i][k],Configs[i][k+3],Configs[j][k+3]];
%p A207724       if col = [0,1,1] or col = [1,0,1] then return 0 fi;
%p A207724    od;
%p A207724    1
%p A207724 end proc:
%p A207724 T:= Matrix(36,36,compat):
%p A207724 u:= Vector[row](36, 1):
%p A207724 v:= Vector(36,1):
%p A207724 6,seq(u . T^(n-2) . v,n=2..50); # _Robert Israel_, Mar 05 2018
%t A207724 LinearRecurrence[{3, -2, 3, -6, 0, 0, 3, 1, 0, -1}, {6, 36, 78, 189, 490, 1113, 2449, 5474, 12036, 26100}, 30] (* _Jean-François Alcover_, May 15 2023, after _Robert Israel_'s confirmed formula *)
%Y A207724 Cf. A207729.
%K A207724 nonn,easy
%O A207724 1,1
%A A207724 _R. H. Hardin_, Feb 19 2012