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T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 1, 3 or 4 king-move neighboring 1s.
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%I #4 Dec 20 2017 22:00:43

%S 1,2,2,3,8,3,4,15,15,4,6,41,47,41,6,9,122,164,164,122,9,13,308,694,

%T 1011,694,308,13,19,855,2605,6747,6747,2605,855,19,28,2405,10149,

%U 40956,77705,40956,10149,2405,28,41,6522,40768,261107,754400,754400,261107,40768

%N T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 1, 3 or 4 king-move neighboring 1s.

%C Table starts

%C ..1....2......3........4.........6...........9............13..............19

%C ..2....8.....15.......41.......122.........308...........855............2405

%C ..3...15.....47......164.......694........2605.........10149...........40768

%C ..4...41....164.....1011......6747.......40956........261107.........1679397

%C ..6..122....694.....6747.....77705......754400.......7978323........85842977

%C ..9..308...2605....40956....754400....11910888.....203660310......3537017110

%C .13..855..10149...261107...7978323...203660310....5712715858....163079979478

%C .19.2405..40768..1679397..85842977..3537017110..163079979478...7684218661876

%C .28.6522.159911.10713116.905014451.60565369990.4582023008983.355189064928041

%H R. H. Hardin, <a href="/A296804/b296804.txt">Table of n, a(n) for n = 1..199</a>

%F Empirical for column k:

%F k=1: a(n) = a(n-1) +a(n-3)

%F k=2: a(n) = a(n-1) +3*a(n-2) +9*a(n-3) -4*a(n-4) -12*a(n-5) -16*a(n-6)

%F k=3: [order 16]

%F k=4: [order 39]

%e Some solutions for n=5 k=4

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

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

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

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

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

%Y Column 1 is A000930(n+1).

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Dec 20 2017