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T(n,k)=Number of nXk 0..3 arrays x(i,j) with each element horizontally or vertically next to at least one element with value (x(i,j)+1) mod 4 and at least one element with value (x(i,j)-1) mod 4, no adjacent elements equal, and upper left element zero
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%I #4 Oct 31 2013 14:44:41

%S 0,0,0,0,2,0,0,2,2,0,0,10,14,10,0,0,22,34,34,22,0,0,70,198,388,198,70,

%T 0,0,186,810,2722,2722,810,186,0,0,538,3706,23884,40762,23884,3706,

%U 538,0,0,1494,16918,190364,579218,579218,190364,16918,1494,0,0,4230,76318

%N T(n,k)=Number of nXk 0..3 arrays x(i,j) with each element horizontally or vertically next to at least one element with value (x(i,j)+1) mod 4 and at least one element with value (x(i,j)-1) mod 4, no adjacent elements equal, and upper left element zero

%C Table starts

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

%C .0...2.....2......10........22..........70..........186............538

%C .0...2....14......34.......198.........810.........3706..........16918

%C .0..10....34.....388......2722.......23884.......190364........1588552

%C .0..22...198....2722.....40762......579218......8554562......124839450

%C .0..70...810...23884....579218....15392914....400106488....10548244824

%C .0.186..3706..190364...8554562...400106488..18831479468...887265863604

%C .0.538.16918.1588552.124839450.10548244824.887265863604.75228804427354

%H R. H. Hardin, <a href="/A230899/b230899.txt">Table of n, a(n) for n = 1..143</a>

%F Empirical for column k:

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

%F k=3: [order 11]

%F k=4: [order 44] for n>45

%e Some solutions for n=4 k=4

%e ..0..1..0..3....0..1..0..3....0..3..2..3....0..3..2..3....0..3..2..3

%e ..3..2..1..2....3..2..1..2....1..0..1..0....1..2..1..0....1..2..1..0

%e ..2..3..2..3....0..3..0..3....2..3..0..3....2..3..0..1....0..3..2..3

%e ..1..0..1..0....1..2..1..0....1..0..1..2....1..0..3..2....1..2..1..0

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Oct 31 2013