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A183821 1/256 the number of (n+1) X 3 0..3 arrays with no 2 X 2 subblock being a reflection across the shared element pair of any horizontal or vertical neighbor.

%I A183821 #16 Aug 12 2018 02:43:48
%S A183821 15,813,43947,2377341,128578815,6954559893,376152548283,
%T A183821 20345104031589,1100412240703119,59518368554767389,
%U A183821 3219189893127084843,174117402720326269485,9417546276713264441151,509369980225226329837125
%N A183821 1/256 the number of (n+1) X 3 0..3 arrays with no 2 X 2 subblock being a reflection across the shared element pair of any horizontal or vertical neighbor.
%C A183821 Column 2 of A183827.
%H A183821 R. H. Hardin, <a href="/A183821/b183821.txt">Table of n, a(n) for n = 1..147</a>
%H A183821 Robert Israel, <a href="/A183821/a183821.pdf">Maple-assisted proof of formula</a>
%F A183821 Empirical: a(n) = 43*a(n-1) + 645*a(n-2) - 2451*a(n-3).
%F A183821 Empirical g.f.: 3*x*(5 + 56*x - 229*x^2) / (1 - 43*x - 645*x^2 + 2451*x^3). - _Colin Barker_, Apr 05 2018
%F A183821 Empirical formulas verified (see link). - _Robert Israel_, Aug 10 2018
%e A183821 Some solutions with upper left block zero for 5 X 3:
%e A183821    0 0 2   0 0 0   0 0 3   0 0 3   0 0 0   0 0 2   0 0 0
%e A183821    0 0 0   0 0 1   0 0 3   0 0 2   0 0 1   0 0 1   0 0 2
%e A183821    2 0 0   2 0 1   3 0 2   1 0 2   1 0 3   1 0 0   3 0 1
%e A183821    2 2 1   3 2 3   1 3 1   1 3 1   0 3 3   2 1 2   0 3 3
%e A183821    3 2 3   0 0 3   0 0 3   2 0 3   3 2 2   0 0 3   2 2 0
%p A183821 Configs:= remove(t -> [t[1],t[4]]=[t[3],t[6]], [seq(convert(4^6+i,base,4)[1..6],i=0..4^6-1)]): nConfigs:= nops(Configs):
%p A183821 Compatible:= proc(i,j) local k;
%p A183821 if Configs[i][4..6] <> Configs[j][1..3] or Configs[i][1..2] =Configs[j][4..5] or Configs[i][2..3] = Configs[j][5..6] then 0 else 1 fi;
%p A183821 end proc:
%p A183821 T:= Matrix(nConfigs,nConfigs,Compatible):
%p A183821 u:= Vector(nConfigs,1):
%p A183821 Tu[0]:= u:
%p A183821 for n from 1 to 30 do Tu[n]:= T . Tu[n-1] od:
%p A183821 seq(u^%T . Tu[n]/256, n=0..30); # _Robert Israel_, Aug 10 2018
%Y A183821 Cf. A183827.
%K A183821 nonn
%O A183821 1,1
%A A183821 _R. H. Hardin_, Jan 07 2011