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T(n,k) = Number of n X k 0..1 arrays with every element equal to 1, 2, 3, 4 or 5 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.
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%I #6 May 27 2021 14:53:22

%S 0,1,0,1,3,0,2,15,11,0,3,46,86,34,0,5,161,519,587,111,0,8,601,3626,

%T 6531,3815,361,0,13,2208,26167,87901,80589,25131,1172,0,21,8053,

%U 185810,1248691,2104533,998670,164916,3809,0,34,29415,1317541,17374552,58679318

%N T(n,k) = Number of n X k 0..1 arrays with every element equal to 1, 2, 3, 4 or 5 horizontally, diagonally or antidiagonally adjacent elements, with upper left element zero.

%C Table starts

%C .0.....1.......1..........2............3...............5..................8

%C .0.....3......15.........46..........161.............601...............2208

%C .0....11......86........519.........3626...........26167.............185810

%C .0....34.....587.......6531........87901.........1248691...........17374552

%C .0...111....3815......80589......2104533........58679318.........1596912288

%C .0...361...25131.....998670.....50519822......2766909379.......147310312318

%C .0..1172..164916...12365841...1212025201....130376252119.....13578993819785

%C .0..3809.1083375..153141597..29081585941...6144174797769...1251888966185979

%C .0.12377.7114906.1896492042.697771332458.289545909430332.115412264434282781

%H R. H. Hardin, <a href="/A302953/b302953.txt">Table of n, a(n) for n = 1..219</a>

%F Empirical for column k:

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

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

%F k=3: a(n) = 4*a(n-1) +15*a(n-2) +13*a(n-3) -2*a(n-4) -19*a(n-5) -3*a(n-6) +4*a(n-8)

%F k=4: [order 13]

%F k=5: [order 43] for n>44

%F Empirical for row n:

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

%F n=2: a(n) = 3*a(n-1) +a(n-2) +4*a(n-3) +4*a(n-4) for n>5

%F n=3: [order 7] for n>9

%F n=4: [order 24] for n>25

%F n=5: [order 73] for n>74

%e Some solutions for n=5, k=4

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

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

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

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

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

%Y Column 2 is A180762.

%Y Row 1 is A000045(n-1).

%Y Row 2 is A232077(n-1).

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Apr 16 2018