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A181256 Number of (n+2)X4 binary matrices with every 3X3 block having exactly four 1's 2

%I #8 Jul 11 2018 04:32:38

%S 336,746,1684,3942,10348,27554,74784,212570,608476,1755630,5152996,

%T 15171626,44814936,133231586,396621844,1182345462,3533450908,

%U 10565961074,31613799504,94685091050,283660310476,850017916350,2548227410836

%N Number of (n+2)X4 binary matrices with every 3X3 block having exactly four 1's

%C Column 2 of A181262

%H R. H. Hardin, <a href="/A181256/b181256.txt">Table of n, a(n) for n=1..200</a>

%H Robert Israel, <a href="/A181256/a181256.pdf">Maple-assisted proof of formula</a>

%F Empirical: a(n)=4*a(n-1)-3*a(n-2)+32*a(n-3)-128*a(n-4)+96*a(n-5)-375*a(n-6)+1500*a(n-7)-1125*a(n-8)+1980*a(n-9)-7920*a(n-10)+5940*a(n-11)-4644*a(n-12)+18576*a(n-13)-13932*a(n-14)+3888*a(n-15)-15552*a(n-16)+11664*a(n-17).

%F Empirical formula verified by _Robert Israel_, Jul 11 2018 (see link).

%e Some solutions for 5X4

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

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

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

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

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

%p okconfig:= proc(L) andmap(i -> L[i+1]+L[i+2]+L[i+3]+L[i+5]+L[i+6]+L[i+7]<=4,[$0..1]) end proc:

%p Configs:= select(okconfig, [seq(convert(2^8+i,base,2)[1..8],i=0..2^8-1)]):

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

%p if Configs[i][1..4] <> Configs[j][5..8] then return 0 fi;

%p if add(Configs[i][k],k=[1,2,3,5,6,7])+add(Configs[j][k],k=1..3)=4 and

%p add(Configs[i][k],k=[2,3,4,6,7,8])+add(Configs[j][k],k=2..4)=4 then 1 else 0 fi

%p end proc:

%p T:= Matrix(213,213,Compatible):

%p v:= Vector(213,1):

%p TV[0]:= v:

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

%p seq(v^%T . TV[n],n=1..30); # _Robert Israel_, Jul 11 2018

%K nonn

%O 1,1

%A _R. H. Hardin_, Oct 10 2010

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)