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T(n,k)=Number of nXk 0..2 arrays with no element equal to more than one of its king-move neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.
8

%I #4 Jan 15 2017 09:31:56

%S 0,0,0,1,0,1,2,24,24,2,8,112,297,112,8,28,504,1046,1046,504,28,94,

%T 2528,3460,2368,3460,2528,94,304,11712,10790,5368,5368,10790,11712,

%U 304,960,52160,32692,12064,9484,12064,32692,52160,960,2976,230048,96454,27218

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

%C Table starts

%C ....0.......0......1......2......8.....28.....94....304.....960....2976....9096

%C ....0.......0.....24....112....504...2528..11712..52160..230048.1000320.4294016

%C ....1......24....297...1046...3460..10790..32692..96454..279958..801574.2269012

%C ....2.....112...1046...2368...5368..12064..27218..61508..137320..303016..661750

%C ....8.....504...3460...5368...9484..17592..33199..63976..124361..236858..440416

%C ...28....2528..10790..12064..17592..29408..51158..90312..164300..299484..534922

%C ...94...11712..32692..27218..33199..51158..85874.144256..244623..428484..747152

%C ..304...52160..96454..61508..63976..90312.144256.239368..392012..649748.1109780

%C ..960..230048.279958.137320.124361.164300.244623.392012..635490.1018244.1657226

%C .2976.1000320.801574.303016.236858.299484.428484.649748.1018244.1614888.2549762

%H R. H. Hardin, <a href="/A281139/b281139.txt">Table of n, a(n) for n = 1..363</a>

%F Empirical for column k:

%F k=1: a(n) = 4*a(n-1) -8*a(n-3) -4*a(n-4) for n>7

%F k=2: [order 8]

%F k=3: [order 14] for n>19

%F k=4: [order 24] for n>33

%F k=5: [order 24] for n>33

%F k=6: [order 32] for n>43

%e Some solutions for n=4 k=4

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

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

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

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

%Y Column 1 is A280279.

%K nonn,tabl

%O 1,7

%A _R. H. Hardin_, Jan 15 2017