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T(n,k)=Number of nXk 0..3 arrays with no element x(i,j) adjacent to value 3-x(i,j) horizontally, diagonally or antidiagonally, top left element zero, and 1 appearing before 2 in row major order
13

%I #4 Dec 04 2013 06:43:14

%S 1,2,3,5,11,10,14,65,74,36,41,386,941,515,136,122,2315,11486,13721,

%T 3602,528,365,13886,141566,342626,200165,25211,2080,1094,83315,

%U 1742447,8714705,10221326,2920145,176474,8256,3281,499886,21452183,221113913

%N T(n,k)=Number of nXk 0..3 arrays with no element x(i,j) adjacent to value 3-x(i,j) horizontally, diagonally or antidiagonally, top left element zero, and 1 appearing before 2 in row major order

%C Table starts

%C ......1........2............5..............14.................41

%C ......3.......11...........65.............386...............2315

%C .....10.......74..........941...........11486.............141566

%C .....36......515........13721..........342626............8714705

%C ....136.....3602.......200165........10221326..........537122150

%C ....528....25211......2920145.......304926626........33113065637

%C ...2080...176474.....42601181......9096692126......2041493495546

%C ...8256..1235315....621496841....271376130626....125863931140721

%C ..32896..8647202...9066845525...8095800458126...7759890074654654

%C .131328.60530411.132273701825.241517133090626.478420800866866973

%H R. H. Hardin, <a href="/A233098/b233098.txt">Table of n, a(n) for n = 1..199</a>

%F Empirical for column k:

%F k=1: a(n) = 6*a(n-1) -8*a(n-2)

%F k=2: a(n) = 8*a(n-1) -7*a(n-2)

%F k=3: a(n) = 16*a(n-1) -21*a(n-2) +6*a(n-3)

%F k=4: a(n) = 31*a(n-1) -35*a(n-2) +5*a(n-3)

%F k=5: [order 11]

%F k=6: [order 22]

%F Empirical for row n:

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

%F n=2: a(n) = 6*a(n-1) +a(n-2) -6*a(n-3) for n>4

%F n=3: a(n) = 13*a(n-1) -5*a(n-2) -47*a(n-3) +52*a(n-4) -12*a(n-5) for n>6

%F n=4: [order 11] for n>12

%F n=5: [order 27] for n>28

%F n=6: [order 87] for n>88

%e Some solutions for n=3 k=4

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

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

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

%Y Column 1 is A007582(n-1)

%Y Column 2 is A199417(n-1)

%Y Row 1 is A007051(n-1)

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Dec 04 2013