A189619 - OEIS

%I #13 Oct 20 2019 15:55:18

%S 16,256,1723,8496,50024,357323,2482591,15915001,100745265,655633882,

%T 4322765564,28292943180,183873191611,1195974189947,7802581300173,

%U 50931021729006,332106212299242,2164490825527781,14110729853099870

%N Number of 4 X n binary arrays without the pattern 0 1 0 diagonally, antidiagonally or horizontally.

%H R. H. Hardin, <a href="/A189619/b189619.txt">Table of n, a(n) for n = 1..200</a>

%H Robert Israel, <a href="/A189619/a189619.pdf">Maple-assisted proof of empirical recurrence</a>

%F Empirical: a(n) = 8*a(n-1) -14*a(n-2) +10*a(n-3) +258*a(n-4) -737*a(n-5) -1502*a(n-6) +1780*a(n-7) +8033*a(n-8) -4800*a(n-9) -30115*a(n-10) +54016*a(n-11) -79554*a(n-12) -215482*a(n-13) +787373*a(n-14) -626468*a(n-15) -426934*a(n-16) +3545678*a(n-17) -6093224*a(n-18) +8409993*a(n-19) -6556066*a(n-20) +986540*a(n-21) -4549030*a(n-22) +9400135*a(n-23) -23776243*a(n-24) +29721357*a(n-25) -36187856*a(n-26) +30281962*a(n-27) -31933555*a(n-28) +10845312*a(n-29) +5786415*a(n-30) +7694508*a(n-31) +5236176*a(n-32) -9921669*a(n-33) -713908*a(n-34) +1520657*a(n-35) -3672439*a(n-36) +1127159*a(n-37) +4759522*a(n-38) -673175*a(n-39) -2818276*a(n-40) -151046*a(n-41) +1043153*a(n-42) +59137*a(n-43) -154754*a(n-44) -24424*a(n-45) +16833*a(n-46) +160*a(n-47) -631*a(n-48) -60*a(n-49) +4*a(n-50).

%F Empirical formula verified (see link). - _Robert Israel_, Oct 20 2019

%e Some solutions for 4 X 3:

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

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

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

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

%p Configs:= [seq(convert(n,base,2)[1..8],n=2^8..2^9-1)]:

%p Compatible:= proc(i,j) local Xi,Xj,k;

%p Xi:= Configs[i]; Xj:= Configs[j];

%p if Xi[5..8] <> Xj[1..4] then return 0 fi;

%p if Xi[1]=0 and ((Xi[5]=1 and Xj[5]=0) or (Xi[6]=1 and Xj[7]=0)) then return 0 fi;

%p if Xi[2]=0 and ((Xi[6]=1 and Xj[6]=0) or (Xi[7]=1 and Xj[8]=0)) then return 0 fi;

%p if Xi[3]=0 and ((Xi[6]=1 and Xj[5]=0) or (Xi[7]=1 and Xj[7]=0)) then return 0 fi;

%p if Xi[4]=0 and ((Xi[7]=1 and Xj[6]=0) or (Xi[8]=1 and Xj[8]=0)) then return 0 fi;

%p 1

%p end proc:

%p T:= Matrix(256,256,Compatible):

%p Tu[0]:= u:

%p for nn from 1 to 30 do Tu[nn]:= T . Tu[nn-1] od:

%p [16, seq(u^%T . Tu[n],n=0..30)]; # _Robert Israel_, Oct 20 2019

%Y Row 4 of A189617.

%K nonn

%O 1,1

%A _R. H. Hardin_, Apr 24 2011