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T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4, 6 or 8 king-move adjacent elements, with upper left element zero.
5

%I #4 Feb 18 2018 10:21:49

%S 1,2,2,4,8,4,8,31,31,8,16,121,179,121,16,32,472,1073,1073,472,32,64,

%T 1841,6479,10150,6479,1841,64,128,7181,39015,97462,97462,39015,7181,

%U 128,256,28010,235033,932318,1502630,932318,235033,28010,256,512,109255,1416220

%N T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 1, 2, 3, 4, 6 or 8 king-move adjacent elements, with upper left element zero.

%C Table starts

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

%C ...2......8......31.......121.........472..........1841............7181

%C ...4.....31.....179......1073........6479.........39015..........235033

%C ...8....121....1073.....10150.......97462........932318.........8918662

%C ..16....472....6479.....97462.....1502630......23034971.......353167830

%C ..32...1841...39015....932318....23034971.....564998177.....13855907809

%C ..64...7181..235033...8918662...353167830...13855907809....543386033722

%C .128..28010.1416220..85379274..5421997554..340475803209..21367169090398

%C .256.109255.8533123.817325435.83244577728.8367066746663.840301228632106

%H R. H. Hardin, <a href="/A299746/b299746.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) = 3*a(n-1) +3*a(n-2) +2*a(n-3)

%F k=3: [order 10] for n>11

%F k=4: [order 37] for n>38

%e Some solutions for n=5 k=4

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

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

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

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

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

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

%Y Column 2 is A281831.

%Y Column 3 is A298996.

%Y Column 4 is A298997.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Feb 18 2018