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T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 4 or 7 king-move adjacent elements, with upper left element zero.
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%I #4 Jan 21 2018 06:12:16

%S 0,1,1,1,4,1,2,17,17,2,3,61,109,61,3,5,216,588,588,216,5,8,793,3276,

%T 4771,3276,793,8,13,2907,18500,41762,41762,18500,2907,13,21,10622,

%U 104034,367315,575754,367315,104034,10622,21,34,38809,585134,3215618,7967553

%N T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 4 or 7 king-move adjacent elements, with upper left element zero.

%C Table starts

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

%C ..1.....4......17........61.........216...........793............2907

%C ..1....17.....109.......588........3276.........18500..........104034

%C ..2....61.....588......4771.......41762........367315.........3215618

%C ..3...216....3276.....41762......575754.......7967553.......109775311

%C ..5...793...18500....367315.....7967553.....173576070......3761355239

%C ..8..2907..104034...3215618...109775311....3761355239....128132389259

%C .13.10622..585134..28178880..1514903717...81677753297...4376310739604

%C .21.38809.3291766.246983338.20908710812.1774108927621.149522986489383

%H R. H. Hardin, <a href="/A298547/b298547.txt">Table of n, a(n) for n = 1..180</a>

%F Empirical for column k:

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

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

%F k=3: [order 12] for n>13

%F k=4: [order 36] for n>39

%e Some solutions for n=6 k=4

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

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

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

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

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

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

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

%Y Column 2 is A297917.

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Jan 21 2018