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T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element horizontally or antidiagonally next to at least one element with value (x(i,j)+1) mod 3, and upper left element zero
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%I #4 Oct 25 2013 13:02:33

%S 0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,8,30,0,0,0,0,30,260,348,0,0,0,0,108,

%T 2358,7072,3956,0,0,0,0,386,20430,149664,186396,44916,0,0,0,0,1376,

%U 176774,3023532,9244260,4899200,509978,0,0,0,0,4902,1527918,61203362

%N T(n,k)=Number of nXk 0..2 arrays x(i,j) with each element horizontally or antidiagonally next to 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 .0.0.0.......2..........8............30..............108................386

%C .0.0.0......30........260..........2358............20430.............176774

%C .0.0.0.....348.......7072........149664..........3023532...........61203362

%C .0.0.0....3956.....186396.......9244260........434369516........20503681466

%C .0.0.0...44916....4899200.....570014364......62312057642......6859445695566

%C .0.0.0..509978..128706852...35145166536....8937075346246...2294186499807542

%C .0.0.0.5790456.3381000336.2166884610178.1281770005911328.767293632652220616

%H R. H. Hardin, <a href="/A230614/b230614.txt">Table of n, a(n) for n = 1..239</a>

%F Empirical for column k:

%F k=4: a(n) = 15*a(n-1) -45*a(n-2) +42*a(n-3) -12*a(n-4) +a(n-5)

%F k=5: a(n) = 31*a(n-1) -126*a(n-2) +42*a(n-3) +79*a(n-4) -a(n-5) +8*a(n-6) for n>7

%F k=6: [order 22] for n>23

%F k=7: [order 39] for n>41

%F Empirical for row n:

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

%F n=3: [order 14] for n>17

%F n=4: [order 59] for n>63

%e Some solutions for n=4 k=4

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

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

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

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

%Y Row 2 is A230269(n-1)

%K nonn,tabl

%O 1,12

%A _R. H. Hardin_, Oct 25 2013