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T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with every element equal to at least one vertical or antidiagonal neighbor and the top left element equal to 0.
6

%I #6 Aug 12 2016 06:55:35

%S 0,0,0,0,1,0,0,1,0,0,0,3,8,6,0,0,7,14,36,0,0,0,17,52,906,456,120,0,0,

%T 42,1516,31818,23592,6360,0,0,0,104,7582,770406,10213152,3304920,

%U 93600,5040,0,0,259,46338,28194516,871718016,4067565720,334738800,1784160,0,0,0,648

%N T(n,k)=Number of nXk arrays containing k copies of 0..n-1 with every element equal to at least one vertical or antidiagonal neighbor and the top left element equal to 0.

%C Table starts

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

%C .0....1.......1.........3..........7............17...........42.......104

%C .0....0.......8........14.........52..........1516.........7582.....46338

%C .0....6......36.......906......31818........770406.....28194516.857204082

%C .0....0.....456.....23592...10213152.....871718016.131259920304

%C .0..120....6360...3304920.4067565720.2310125813160

%C .0....0...93600.334738800

%C .0.5040.1784160

%C .0....0

%C .0

%H R. H. Hardin, <a href="/A268159/b268159.txt">Table of n, a(n) for n = 1..61</a>

%e Some solutions for n=4 k=4

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

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

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

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

%Y Column 2 is A005212(n-1).

%Y Row 2 is A003440(n-2).

%K nonn,tabl

%O 1,12

%A _R. H. Hardin_, Jan 27 2016