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T(n,k)=Number of nXk 0..1 arrays with every element unequal to 0, 2, 3 or 5 king-move adjacent elements, with upper left element zero.
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%I #4 May 04 2018 11:22:43

%S 1,1,1,1,4,1,1,7,7,1,1,13,13,13,1,1,26,21,21,26,1,1,49,27,26,27,49,1,

%T 1,99,53,64,64,53,99,1,1,194,99,115,137,115,99,194,1,1,387,197,211,

%U 271,271,211,197,387,1,1,773,371,439,655,538,655,439,371,773,1,1,1538,713,870

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

%C Table starts

%C .1...1...1...1....1....1.....1.....1......1......1.......1.......1........1

%C .1...4...7..13...26...49....99...194....387....773....1538....3081.....6147

%C .1...7..13..21...27...53....99...197....371....713....1365....2645.....5105

%C .1..13..21..26...64..115...211...439....870...1725....3513....7141....14372

%C .1..26..27..64..137..271...655..1412...3245...7809...17290...40682....93292

%C .1..49..53.115..271..538..1458..2973...7826..19113...47074..115148...283156

%C .1..99..99.211..655.1458..4224.10633..28365..75175..201031..544217..1452266

%C .1.194.197.439.1412.2973.10633.30468..84398.254233..751747.2193664..6512888

%C .1.387.371.870.3245.7826.28365.84398.264104.857398.2706019.8591303.27513653

%H R. H. Hardin, <a href="/A304010/b304010.txt">Table of n, a(n) for n = 1..264</a>

%F Empirical for column k:

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

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

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

%F k=4: [order 67] for n>70

%e Some solutions for n=5 k=4

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

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

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

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

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

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, May 04 2018