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T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 4, 5, 7 or 8 king-move adjacent elements, with upper left element zero.
7

%I #4 Feb 21 2018 11:51:55

%S 1,2,2,4,7,4,8,13,13,8,16,29,20,29,16,32,73,41,41,73,32,64,157,102,

%T 125,102,157,64,128,353,253,585,585,253,353,128,256,869,632,1974,3068,

%U 1974,632,869,256,512,1993,1563,6542,11263,11263,6542,1563,1993,512,1024,4557

%N T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 4, 5, 7 or 8 king-move adjacent elements, with upper left element zero.

%C Table starts

%C ...1....2....4.....8......16.......32........64........128.........256

%C ...2....7...13....29......73......157.......353........869........1993

%C ...4...13...20....41.....102......253.......632.......1563........3896

%C ...8...29...41...125.....585.....1974......6542......24545.......89192

%C ..16...73..102...585....3068....11263.....53159.....246537.....1098723

%C ..32..157..253..1974...11263....54005....324797....1859944....10716128

%C ..64..353..632..6542...53159...324797...2714797...21236116...164681059

%C .128..869.1563.24545..246537..1859944..21236116..213883244..2166043717

%C .256.1993.3896.89192.1098723.10716128.164681059.2166043717.29310012663

%H R. H. Hardin, <a href="/A299879/b299879.txt">Table of n, a(n) for n = 1..180</a>

%F Empirical for column k:

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

%F k=2: a(n) = 4*a(n-1) -5*a(n-2) +10*a(n-3) -24*a(n-4) +16*a(n-5) for n>6

%F k=3: [order 16] for n>17

%F k=4: [order 67] for n>68

%e Some solutions for n=5 k=4

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

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

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

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

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

%Y Column 1 is A000079(n-1).

%Y Column 2 is A298215.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Feb 21 2018