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T(n,k)=Number of nXk 0..1 arrays with every element unequal to 2, 3 or 4 horizontally, vertically or antidiagonally adjacent elements, with upper left element zero.
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%I #4 Aug 10 2018 10:33:38

%S 0,0,0,0,1,0,0,1,1,0,0,2,4,2,0,0,5,13,13,5,0,0,16,59,82,59,16,0,0,45,

%T 252,549,549,252,45,0,0,123,1074,3697,6193,3697,1074,123,0,0,340,4633,

%U 25169,67792,67792,25169,4633,340,0,0,946,19933,170608,741031,1184196,741031

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

%C Table starts

%C .0...0.....0.......0........0..........0............0..............0

%C .0...1.....1.......2........5.........16...........45............123

%C .0...1.....4......13.......59........252.........1074...........4633

%C .0...2....13......82......549.......3697........25169.........170608

%C .0...5....59.....549.....6193......67792.......741031........8119416

%C .0..16...252....3697....67792....1184196.....20734591......364026326

%C .0..45..1074...25169...741031...20734591....580702029....16327953218

%C .0.123..4633..170608..8119416..364026326..16327953218...736219004144

%C .0.340.19933.1156219.88900423.6382316173.458323749120.33124631682770

%H R. H. Hardin, <a href="/A317896/b317896.txt">Table of n, a(n) for n = 1..220</a>

%F Empirical for column k:

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

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

%F k=3: [order 16] for n>18

%F k=4: [order 54] for n>56

%e Some solutions for n=5 k=4

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

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

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

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

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

%K nonn,tabl

%O 1,12

%A _R. H. Hardin_, Aug 10 2018