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T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero.
6

%I #4 Feb 06 2018 13:23:26

%S 0,1,1,1,4,1,2,18,18,2,3,52,56,52,3,5,174,223,223,174,5,8,604,849,

%T 1024,849,604,8,13,2048,3387,5360,5360,3387,2048,13,21,6948,13075,

%U 28374,55390,28374,13075,6948,21,34,23652,51006,144040,482418,482418,144040,51006

%N T(n,k)=Number of nXk 0..1 arrays with every element equal to 1, 2, 3, 5, 6 or 7 king-move adjacent elements, with upper left element zero.

%C Table starts

%C ..0.....1......1.......2.........3...........5............8.............13

%C ..1.....4.....18......52.......174.........604.........2048...........6948

%C ..1....18.....56.....223.......849........3387........13075..........51006

%C ..2....52....223....1024......5360.......28374.......144040.........743640

%C ..3...174....849....5360.....55390......482418......3860858.......34734150

%C ..5...604...3387...28374....482418.....6525374.....76611604.....1041660250

%C ..8..2048..13075..144040...3860858....76611604...1227812226....23819382545

%C .13..6948..51006..743640..34734150..1041660250..23819382545...723886147451

%C .21.23652.199243.3857242.308990628.14216373088.471369052777.22415064379413

%H R. H. Hardin, <a href="/A299307/b299307.txt">Table of n, a(n) for n = 1..180</a>

%F Empirical for column k:

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

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

%F k=3: [order 20] for n>21

%F k=4: [order 68] for n>70

%e Some solutions for n=5 k=4

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

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

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

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

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

%Y Column 1 is A000045(n-1).

%Y Column 2 is A297945.

%Y Column 3 is A298384.

%K nonn,tabl

%O 1,5

%A _R. H. Hardin_, Feb 06 2018