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T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1, 2, 3, 4, 6, 7 or 8 king-move adjacent elements, with upper left element zero.
7

%I #4 Jul 31 2018 12:46:57

%S 1,2,2,4,8,4,8,30,30,8,16,112,170,112,16,32,420,997,997,420,32,64,

%T 1576,5837,9513,5837,1576,64,128,5912,34152,91461,91461,34152,5912,

%U 128,256,22176,199987,869466,1489357,869466,199987,22176,256,512,83184,1170900

%N T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1, 2, 3, 4, 6, 7 or 8 king-move adjacent elements, with upper left element zero.

%C Table starts

%C ...1.....2.......4.........8..........16.............32...............64

%C ...2.....8......30.......112.........420...........1576.............5912

%C ...4....30.....170.......997........5837..........34152...........199987

%C ...8...112.....997......9513.......91461.........869466..........8305672

%C ..16...420....5837.....91461.....1489357.......23570355........376898848

%C ..32..1576...34152....869466....23570355......611929512......16121282196

%C ..64..5912..199987...8305672...376898848....16121282196.....703696012516

%C .128.22176.1170900..79320881..6024751866...424656152144...30722175114871

%C .256.83184.6855565.757097096.96177867099.11163587715218.1337502015395808

%H R. H. Hardin, <a href="/A317572/b317572.txt">Table of n, a(n) for n = 1..180</a>

%F Empirical for column k:

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

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

%F k=3: [order 11] for n>12

%F k=4: [order 38] for n>39

%e Some solutions for n=5 k=4

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

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

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

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

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

%Y Column 1 is A000079(n-1).

%Y Column 2 is A281949.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Jul 31 2018