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T(n,k)=Number of nXk 0..2 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly two elements, and with new values introduced in order 0 sequentially upwards.
11

%I #4 Jan 10 2017 10:25:46

%S 0,0,0,0,1,0,1,10,9,0,2,213,646,124,0,9,2292,22568,22632,1464,0,34,

%T 21762,492490,1451655,610448,15768,0,124,184076,9426050,65348136,

%U 75809243,14262832,159920,0,432,1457827,162640161,2571528867,7083739466

%N T(n,k)=Number of nXk 0..2 arrays with no element equal to more than one of its horizontal and antidiagonal neighbors, with the exception of exactly two elements, and with new values introduced in order 0 sequentially upwards.

%C Table starts

%C .0........0............0...............1..................2...................9

%C .0........1...........10.............213...............2292...............21762

%C .0........9..........646...........22568.............492490.............9426050

%C .0......124........22632.........1451655...........65348136..........2571528867

%C .0.....1464.......610448........75809243.........7083739466........574982226478

%C .0....15768.....14262832......3521886844.......684011230518.....114677717497532

%C .0...159920....304584096....151803173493.....61277484218852...21239418911643829

%C .0..1554304...6117000704...6210239609889...5208435552362140.3734730743379447607

%C .0.14632704.117496694272.244458357395448.425813044528570428

%H R. H. Hardin, <a href="/A280902/b280902.txt">Table of n, a(n) for n = 1..96</a>

%F Empirical for column k:

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

%F k=2: [order 6] for n>8

%F k=3: [order 6] for n>8

%F k=4: [order 12] for n>14

%F k=5: [order 24] for n>27

%F Empirical for row n:

%F n=1: a(n) = 6*a(n-1) -6*a(n-2) -16*a(n-3) +12*a(n-4) +24*a(n-5) +8*a(n-6) for n>10

%F n=2: [order 15] for n>17

%F n=3: [order 54] for n>60

%e Some solutions for n=3 k=4

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

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

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

%Y Row 1 is A280309.

%K nonn,tabl

%O 1,8

%A _R. H. Hardin_, Jan 10 2017