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T(n,k)=Number of nXk 0..1 arrays with every element equal to 2, 3, 4 or 6 king-move adjacent elements, with upper left element zero.
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%I #4 Jan 11 2018 08:23:54

%S 0,0,0,0,1,0,0,2,2,0,0,5,4,5,0,0,10,13,13,10,0,0,25,63,59,63,25,0,0,

%T 54,253,346,346,253,54,0,0,125,953,2197,3500,2197,953,125,0,0,282,

%U 3802,13350,33869,33869,13350,3802,282,0,0,641,15108,84657,302143,532282

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

%C Table starts

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

%C .0...1.....2......5.......10.........25...........54...........125

%C .0...2.....4.....13.......63........253..........953..........3802

%C .0...5....13.....59......346.......2197........13350.........84657

%C .0..10....63....346.....3500......33869.......302143.......2864336

%C .0..25...253...2197....33869.....532282......7794251.....117272307

%C .0..54...953..13350...302143....7794251....180902016....4321153141

%C .0.125..3802..84657..2864336..117272307...4321153141..162145163369

%C .0.282.15108.529728.26751944.1752542571.102552409457.6062432627760

%H R. H. Hardin, <a href="/A298070/b298070.txt">Table of n, a(n) for n = 1..180</a>

%F Empirical for column k:

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

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

%F k=3: [order 17]

%F k=4: [order 62] for n>63

%e Some solutions for n=5 k=4

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

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

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

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

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

%Y Column 2 is A297860.

%K nonn,tabl

%O 1,8

%A _R. H. Hardin_, Jan 11 2018