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T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 3 or 4 king-move neighboring 1s.
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%I #4 Dec 04 2017 20:35:46

%S 1,1,1,1,2,1,1,3,3,1,1,4,6,4,1,1,8,9,9,8,1,1,17,26,16,26,17,1,1,31,74,

%T 64,64,74,31,1,1,71,163,185,503,185,163,71,1,1,166,535,528,2800,2800,

%U 528,535,166,1,1,365,1735,2109,14057,29003,14057,2109,1735,365,1,1,856,4960

%N T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 3 or 4 king-move neighboring 1s.

%C Table starts

%C .1...1....1....1......1........1.........1...........1.............1

%C .1...2....3....4......8.......17........31..........71...........166

%C .1...3....6....9.....26.......74.......163.........535..........1735

%C .1...4....9...16.....64......185.......528........2109..........7336

%C .1...8...26...64....503.....2800.....14057......106920........726414

%C .1..17...74..185...2800....29003....197661.....2401049......28504534

%C .1..31..163..528..14057...197661...1993970....34228975.....553395326

%C .1..71..535.2109.106920..2401049..34228975..1061506422...28707285600

%C .1.166.1735.7336.726414.28504534.553395326.28707285600.1365762093736

%H R. H. Hardin, <a href="/A296115/b296115.txt">Table of n, a(n) for n = 1..199</a>

%F Empirical for column k:

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

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

%F k=3: [order 11]

%F k=4: [order 27]

%F k=5: [order 89]

%e Some solutions for n=6 k=4

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

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

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

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

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

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

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Dec 04 2017