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T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 2, 3 or 6 king-move adjacent elements, with upper left element zero.
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%I #4 Jan 10 2018 07:23:22

%S 1,1,1,1,5,1,1,6,6,1,1,17,2,17,1,1,24,5,5,24,1,1,53,7,15,7,53,1,1,94,

%T 14,23,23,14,94,1,1,173,21,43,34,43,21,173,1,1,340,41,79,61,61,79,41,

%U 340,1,1,601,70,184,105,131,105,184,70,601,1,1,1178,129,380,222,242,242,222

%N T(n,k)=Number of nXk 0..1 arrays with every element equal to 0, 2, 3 or 6 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......1......1

%C .1...5..6..17..24...53...94..173..340...601..1178..2137...4056...7565..14086

%C .1...6..2...5...7...14...21...41...70...129...233...428....783...1445...2664

%C .1..17..5..15..23...43...79..184..380...830..1776..3815...8227..17737..38335

%C .1..24..7..23..34...61..105..222..428...948..1975..4162...8801..18972..40777

%C .1..53.14..43..61..131..242..530.1226..2701..5847.13187..30316..68963.156767

%C .1..94.21..79.105..242..444..935.1835..3921..8793.19694..44948.102048.235955

%C .1.173.41.184.222..530..935.1980.3832..8355.18941.40962..89009.197337.448014

%C .1.340.70.380.428.1226.1835.3832.6500.15690.33619.75185.159728.361567.821919

%H R. H. Hardin, <a href="/A297986/b297986.txt">Table of n, a(n) for n = 1..511</a>

%F Empirical for column k:

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

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

%F k=3: [order 11] for n>13

%F k=4: [order 22] for n>28

%F k=5: [order 37] for n>43

%F k=6: [order 76] for n>84

%e Some solutions for n=7 k=4

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

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

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

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

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

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

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

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Jan 10 2018