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T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero.
7

%I #4 Jun 18 2018 07:50:53

%S 1,2,2,4,4,4,8,12,12,8,16,24,27,24,16,32,64,58,58,64,32,64,184,189,

%T 164,189,184,64,128,432,526,608,608,526,432,128,256,1088,1344,2168,

%U 3572,2168,1344,1088,256,512,2944,3772,6955,16833,16833,6955,3772,2944,512,1024

%N T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1, 2, 4, 6 or 7 king-move adjacent elements, with upper left element zero.

%C Table starts

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

%C ...2....4....12....24......64......184.......432.......1088........2944

%C ...4...12....27....58.....189......526......1344.......3772.......10515

%C ...8...24....58...164.....608.....2168......6955......23269.......80463

%C ..16...64...189...608....3572....16833.....67791.....313348.....1479005

%C ..32..184...526..2168...16833....99928....513811....3086539....18841620

%C ..64..432..1344..6955...67791...513811...3363541...25699369...199805626

%C .128.1088..3772.23269..313348..3086539..25699369..257823074..2641827240

%C .256.2944.10515.80463.1479005.18841620.199805626.2641827240.35799252324

%H R. H. Hardin, <a href="/A306060/b306060.txt">Table of n, a(n) for n = 1..219</a>

%F Empirical for column k:

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

%F k=2: a(n) = 2*a(n-1) +8*a(n-3) -8*a(n-4) -8*a(n-5) for n>6

%F k=3: [order 13] for n>14

%F k=4: [order 65] for n>67

%e Some solutions for n=5 k=4

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

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

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

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

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

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

%Y Column 2 is A303794.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Jun 18 2018