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

%I #4 Apr 12 2018 11:21:48

%S 1,2,2,4,8,4,8,32,32,8,16,128,228,128,16,32,512,1637,1652,512,32,64,

%T 2048,11814,21625,11980,2048,64,128,8192,85268,285613,286631,86916,

%U 8192,128,256,32768,615589,3778433,6947036,3798398,630604,32768,256,512,131072

%N T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3 or 4 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......228.......1637.........11814...........85268

%C ...8....128.....1652......21625........285613.........3778433

%C ..16....512....11980.....286631.......6947036.......168799572

%C ..32...2048....86916....3798398.....168833401......7530280825

%C ..64...8192...630604...50347423....4104946296....336148647504

%C .128..32768..4575332..667361051...99807877377..15005851329729

%C .256.131072.33196332.8845980434.2426739457531.669868217032865

%H R. H. Hardin, <a href="/A302741/b302741.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) = 7*a(n-1) +2*a(n-2) +2*a(n-3) -20*a(n-4) -16*a(n-5)

%F k=4: [order 15]

%F k=5: [order 46]

%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) = 7*a(n-1) +4*a(n-2) -17*a(n-3) -3*a(n-4) -9*a(n-6) +14*a(n-7) for n>8

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

%F n=5: [order 64] for n>65

%e Some solutions for n=5 k=4

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

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

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

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

%e ..0..0..1..0. .1..1..0..1. .0..0..1..1. .0..1..1..1. .1..1..0..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 12 2018