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T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 0, 2 or 3 king-move neighboring 1s.
8

%I #8 Jun 19 2020 04:02:33

%S 2,3,3,5,10,5,8,25,25,8,13,68,82,68,13,21,208,334,334,208,21,34,609,

%T 1451,2295,1451,609,34,55,1785,5938,16541,16541,5938,1785,55,89,5375,

%U 25144,110036,213800,110036,25144,5375,89,144,16174,108174,775010,2358963

%N T(n,k)=Number of nXk 0..1 arrays with each 1 adjacent to 0, 2 or 3 king-move neighboring 1s.

%H R. H. Hardin, <a href="/A296674/b296674.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-2)

%F k=2: a(n) = 3*a(n-1) +5*a(n-3) -10*a(n-4) -12*a(n-5) -4*a(n-6)

%F k=3: [order 12]

%F k=4: [order 27]

%F k=5: [order 67]

%e Table starts

%e ..2.....3......5........8.........13...........21.............34

%e ..3....10.....25.......68........208..........609...........1785

%e ..5....25.....82......334.......1451.........5938..........25144

%e ..8....68....334.....2295......16541.......110036.........775010

%e .13...208...1451....16541.....213800......2358963.......28341757

%e .21...609...5938...110036....2358963.....40937523......794602323

%e .34..1785..25144...775010...28341757....794602323....25499216805

%e .55..5375.108174..5466936..346828018..15702927104...833656369935

%e .89.16174.463106.38297415.4150012532.301022037125.26222808591554

%e ...

%e Some solutions for n=4 k=4

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

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

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

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

%Y Column 1 is A000045(n+2).

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_, Dec 18 2017