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T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 4 or 6 king-move adjacent elements, with upper left element zero.
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%I #4 Jan 17 2018 08:24:06

%S 0,1,1,1,4,1,2,17,17,2,3,61,113,61,3,5,216,628,628,216,5,8,793,3641,

%T 5663,3641,793,8,13,2907,21375,51588,51588,21375,2907,13,21,10622,

%U 124972,479767,755110,479767,124972,10622,21,34,38809,730509,4442111,11244900

%N T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 4 or 6 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.....113.......628........3641.........21375..........124972

%C ..2....61.....628......5663.......51588........479767.........4442111

%C ..3...216....3641.....51588......755110......11244900.......166636876

%C ..5...793...21375....479767....11244900.....268494008......6373591375

%C ..8..2907..124972...4442111...166636876....6373591375....242019919082

%C .13.10622..730509..41117169..2469362361..151336485128...9195408207604

%C .21.38809.4271331.380674402.36605480121.3595159432175.349596064308984

%H R. H. Hardin, <a href="/A298337/b298337.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 13] for n>15

%F k=4: [order 48] for n>50

%e Some solutions for n=4 k=4

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

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

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

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

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

%Y Column 2 is A297917.

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Jan 17 2018