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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, and top left element zero
13

%I #4 Dec 02 2013 18:52:42

%S 1,3,4,9,21,16,27,129,147,64,81,771,1881,1029,256,243,4629,22971,

%T 27441,7203,1024,729,27771,283131,685251,400329,50421,4096,2187,

%U 166629,3484893,17429409,20442651,5840289,352947,16384,6561,999771,42904365,442227825

%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, and top left element zero

%C Table starts

%C ......1.........3............9..............27.................81

%C ......4........21..........129.............771...............4629

%C .....16.......147.........1881...........22971.............283131

%C .....64......1029........27441..........685251...........17429409

%C ....256......7203.......400329........20442651.........1074244299

%C ...1024.....50421......5840289.......609853251........66226131273

%C ...4096....352947.....85202361.....18193384251......4082986991091

%C ..16384...2470629...1242993681....542752261251....251727862281441

%C ..65536..17294403..18133691049..16191600916251..15519780149309307

%C .262144.121060821.264547403649.483034266181251.956841601733733945

%H R. H. Hardin, <a href="/A232955/b232955.txt">Table of n, a(n) for n = 1..199</a>

%F Empirical for column k:

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

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

%F k=3: a(n) = 15*a(n-1) -6*a(n-2)

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

%F k=5: [order 10]

%F k=6: [order 21]

%F Empirical for row n:

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

%F n=2: a(n) = 5*a(n-1) +6*a(n-2)

%F n=3: a(n) = 12*a(n-1) +7*a(n-2) -40*a(n-3) +12*a(n-4)

%F n=4: [order 10]

%F n=5: [order 26] for n>27

%F n=6: [order 86] for n>87

%e Some solutions for n=3 k=4

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

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

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

%Y Column 1 is A000302(n-1)

%Y Column 2 is A169634(n-1)

%Y Row 1 is A000244(n-1)

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Dec 02 2013