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T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element horizontally or vertically next to at least one element with value (x(i,j)+1) mod 3 and at least one element with value (x(i,j)-1) mod 3, and upper left element zero
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%I #4 Oct 30 2013 17:09:07

%S 0,0,0,0,0,0,0,2,2,0,0,0,4,0,0,0,2,12,12,2,0,0,12,100,174,100,12,0,0,

%T 18,326,816,816,326,18,0,0,56,2114,12210,28772,12210,2114,56,0,0,170,

%U 9226,108006,497166,497166,108006,9226,170,0,0,420,47220,1202184,10872238

%N T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element horizontally or vertically next to at least one element with value (x(i,j)+1) mod 3 and at least one element with value (x(i,j)-1) mod 3, and upper left element zero

%C Table starts

%C .0..0....0.......0.........0...........0.............0.............0

%C .0..0....2.......0.........2..........12............18............56

%C .0..2....4......12.......100.........326..........2114..........9226

%C .0..0...12.....174.......816.......12210........108006.......1202184

%C .0..2..100.....816.....28772......497166......10872238.....222626294

%C .0.12..326...12210....497166....21835516.....888633610...37511690680

%C .0.18.2114..108006..10872238...888633610...75341251814.6484756891722

%C .0.56.9226.1202184.222626294.37511690680.6484756891722

%H R. H. Hardin, <a href="/A230819/b230819.txt">Table of n, a(n) for n = 1..112</a>

%F Empirical for column k:

%F k=2: a(n) = 2*a(n-1) +a(n-2) +4*a(n-3) -4*a(n-4)

%F k=3: [order 19]

%F k=4: [order 95] for n>96

%e Some solutions for n=4 k=4

%e ..0..2..1..0....0..2..1..0....0..1..2..1....0..1..0..2....0..1..2..0

%e ..1..0..0..2....1..0..1..2....2..1..0..0....2..2..0..1....2..0..2..1

%e ..2..0..1..1....2..0..0..0....0..1..2..2....1..0..1..2....1..2..0..2

%e ..0..1..2..0....0..1..2..1....2..1..0..1....0..2..1..0....0..2..1..0

%K nonn,tabl

%O 1,8

%A _R. H. Hardin_, Oct 30 2013