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

%I #4 Jun 05 2018 18:30:59

%S 1,2,2,4,4,4,8,3,3,8,16,5,6,5,16,32,8,8,8,8,32,64,13,19,15,19,13,64,

%T 128,21,33,38,38,33,21,128,256,34,60,102,76,102,60,34,256,512,55,112,

%U 247,327,327,247,112,55,512,1024,89,205,617,967,1897,967,617,205,89,1024,2048

%N T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 1, 2, 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........512........1024

%C ...2..4...3....5....8.....13......21.......34........55.........89.........144

%C ...4..3...6....8...19.....33......60......112.......205........377.........694

%C ...8..5...8...15...38....102.....247......617......1559.......3892........9739

%C ..16..8..19...38...76....327.....967.....2641......8578......26437.......78868

%C ..32.13..33..102..327...1897....8182....33142....152197.....674090.....2919193

%C ..64.21..60..247..967...8182...50618...280791...1818586...11395966....69071165

%C .128.34.112..617.2641..33142..280791..1944635..17050251..144099899..1144040510

%C .256.55.205.1559.8578.152197.1818586.17050251.206386330.2422285068.26433099143

%H R. H. Hardin, <a href="/A305586/b305586.txt">Table of n, a(n) for n = 1..420</a>

%F Empirical for column k:

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

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

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

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

%F k=5: a(n) = a(n-1) +3*a(n-2) +12*a(n-3) -9*a(n-5) -27*a(n-6) for n>10

%F k=6: [order 15] for n>20

%F k=7: [order 45] for n>50

%e Some solutions for n=5 k=4

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

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

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

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

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

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

%Y Column 2 is A000045(n+2) for n>2.

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Jun 05 2018