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T(n,k)=Number of nXk 0..1 arrays with each 1 horizontally, vertically or antidiagonally adjacent to 1 or 2 neighboring 1s.
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%I #4 Dec 11 2017 11:31:24

%S 1,2,2,4,10,4,7,28,28,7,12,86,127,86,12,21,279,641,641,279,21,37,869,

%T 3237,5389,3237,869,37,65,2728,16248,46786,46786,16248,2728,65,114,

%U 8596,81661,396806,684894,396806,81661,8596,114,200,27004,410199,3372222

%N T(n,k)=Number of nXk 0..1 arrays with each 1 horizontally, vertically or antidiagonally adjacent to 1 or 2 neighboring 1s.

%C Table starts

%C ...1.....2.......4.........7..........12............21..............37

%C ...2....10......28........86.........279...........869............2728

%C ...4....28.....127.......641........3237.........16248...........81661

%C ...7....86.....641......5389.......46786........396806.........3372222

%C ..12...279....3237.....46786......684894.......9703136.......138882856

%C ..21...869...16248....396806.....9703136.....229596763......5498685275

%C ..37..2728...81661...3372222...138882856....5498685275....221204933878

%C ..65..8596..410199..28724880..1988460073..131626644289...8887761158698

%C .114.27004.2061212.244493344.28434977556.3147315906687.356576134627608

%H R. H. Hardin, <a href="/A296386/b296386.txt">Table of n, a(n) for n = 1..337</a>

%F Empirical for column k:

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

%F k=2: a(n) = 3*a(n-1) -a(n-2) +6*a(n-3) -6*a(n-4) +6*a(n-5) -5*a(n-6) +3*a(n-7) -a(n-8)

%F k=3: [order 16]

%F k=4: [order 40]

%F k=5: [order 92]

%e Some solutions for n=5 k=4

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

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

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

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

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

%Y Column 1 is A005251(n+2).

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Dec 11 2017