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T(n,k) = Number of n X k 0..1 arrays with no element equal to more than two of its horizontal, vertical and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.
8

%I #6 Jun 26 2022 20:59:39

%S 0,0,0,0,0,0,0,6,6,0,0,24,55,24,0,0,92,244,244,92,0,0,318,894,958,894,

%T 318,0,0,1056,2948,3330,3330,2948,1056,0,0,3406,9188,10474,10954,

%U 10474,9188,3406,0,0,10770,27580,30808,32244,32244,30808,27580,10770,0,0,33542

%N T(n,k) = Number of n X k 0..1 arrays with no element equal to more than two of its horizontal, vertical and antidiagonal neighbors, with the exception of exactly one element, and with new values introduced in order 0 sequentially upwards.

%C Table starts

%C .0.....0......0......0.......0.......0.......0........0........0........0

%C .0.....0......6.....24......92.....318....1056.....3406....10770....33542

%C .0.....6.....55....244.....894....2948....9188....27580....80579...230740

%C .0....24....244....958....3330...10474...30808....86860...237624...636256

%C .0....92....894...3330...10954...32244...89147...236332...609693..1542674

%C .0...318...2948..10474...32244...89042..231146...578810..1415986..3412832

%C .0..1056...9188..30808...89147..231146..566178..1343304..3132736..7230554

%C .0..3406..27580..86860..236332..578810.1343304..3035844..6776640.15052898

%C .0.10770..80579.237624..609693.1415986.3132736..6776640.14551996.31239986

%C .0.33542.230740.636256.1542674.3412832.7230554.15052898.31239986.65113158

%H R. H. Hardin, <a href="/A281080/b281080.txt">Table of n, a(n) for n = 1..363</a>

%F Empirical for column k:

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

%F k=2: [order 12];

%F k=3: [order 12] for n>16;

%F k=4: [order 19] for n>25;

%F k=5: [order 25] for n>32;

%F k=6: [order 29] for n>37;

%F k=7: [order 33] for n>42.

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

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

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

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

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

%K nonn,tabl

%O 1,8

%A _R. H. Hardin_, Jan 14 2017