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T(n,k) = Number of n X k 0..1 arrays with each 1 horizontally, vertically or antidiagonally adjacent to 1, 2 or 3 neighboring 1's.
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%I #10 Dec 29 2023 23:11:46

%S 1,2,2,4,11,4,7,39,39,7,12,136,228,136,12,21,494,1441,1441,494,21,37,

%T 1785,9194,16524,9194,1785,37,65,6432,58251,194743,194743,58251,6432,

%U 65,114,23201,370000,2271055,4254377,2271055,370000,23201,114,200,83695

%N T(n,k) = Number of n X k 0..1 arrays with each 1 horizontally, vertically or antidiagonally adjacent to 1, 2 or 3 neighboring 1's.

%H R. H. Hardin, <a href="/A296599/b296599.txt">Table of n, a(n) for n = 1..312</a>

%F Empirical for column k:

%F k=1: a(n) = 2*a(n-1) -a(n-2) +a(n-3)

%F k=2: a(n) = 3*a(n-1) +a(n-2) +4*a(n-3) +a(n-4) +a(n-6) -a(n-7) -a(n-8)

%F k=3: [order 17]

%F k=4: [order 44]

%e Table starts

%e ...1.....2........4..........7...........12..............21................37

%e ...2....11.......39........136..........494............1785..............6432

%e ...4....39......228.......1441.........9194...........58251............370000

%e ...7...136.....1441......16524.......194743.........2271055..........26487173

%e ..12...494.....9194.....194743......4254377........91004329........1952733110

%e ..21..1785....58251....2271055.....91004329......3553672817......139547279602

%e ..37..6432...370000...26487173...1952733110....139547279602....10046940260800

%e ..65.23201..2348956..309059200..41935988643...5483284264255...723805695940277

%e .114.83695.14911425.3606151826.900218672974.215325152512080.52099265384617231

%e ...

%e Some solutions for n=4, k=4

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

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

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

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

%Y Column 1 is A005251(n+2).

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Dec 16 2017