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T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.
12

%I #4 Apr 24 2018 10:22:23

%S 1,2,2,4,8,4,8,32,32,8,16,128,232,128,16,32,512,1690,1696,512,32,64,

%T 2048,12340,22756,12408,2048,64,128,8192,90112,306448,306767,90800,

%U 8192,128,256,32768,658204,4129588,7626768,4136339,664512,32768,256,512,131072

%N T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4 or 6 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

%C Table starts

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

%C ...2......8.......32.........128...........512............2048

%C ...4.....32......232........1690.........12340...........90112

%C ...8....128.....1696.......22756........306448.........4129588

%C ..16....512....12408......306767.......7626768.......189848373

%C ..32...2048....90800.....4136339.....189837638......8727953509

%C ..64...8192...664512....55781418....4726484016....401405461699

%C .128..32768..4863312...752277525..117683035940..18461936404456

%C .256.131072.35593024.10145443043.2930192820802.849140799884830

%H R. H. Hardin, <a href="/A303456/b303456.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)

%F k=3: a(n) = 8*a(n-1) -4*a(n-2) -2*a(n-3) -36*a(n-4) -16*a(n-5)

%F k=4: [order 15]

%F k=5: [order 47]

%F Empirical for row n:

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

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

%F n=3: a(n) = 8*a(n-1) -40*a(n-3) +20*a(n-4) +8*a(n-5) -3*a(n-6) +32*a(n-7) for n>8

%F n=4: [order 15] for n>16

%F n=5: [order 68] for n>69

%e Some solutions for n=5 k=4

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

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

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

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

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

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

%Y Column 2 is A004171(n-1).

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

%Y Row 2 is A004171(n-1).

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Apr 24 2018