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T(n,k)=Number of nXk 0..1 arrays with no element equal to more than two of its horizontal, vertical and antidiagonal neighbors, with the exception of exactly two elements, and with new values introduced in order 0 sequentially upwards.
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%I #4 Jan 08 2017 09:47:30

%S 0,0,0,0,1,0,0,3,3,0,0,18,56,18,0,0,72,332,332,72,0,0,284,1578,2146,

%T 1578,284,0,0,1047,6540,9715,9715,6540,1047,0,0,3722,24490,37644,

%U 41931,37644,24490,3722,0,0,12816,86010,133067,155560,155560,133067,86010,12816,0,0

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

%C Table starts

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

%C .0.....1......3......18.......72......284......1047......3722.....12816

%C .0.....3.....56.....332.....1578.....6540.....24490.....86010....288426

%C .0....18....332....2146.....9715....37644....133067....440282...1387898

%C .0....72...1578....9715....41931...155560....521862...1635481...4878157

%C .0...284...6540...37644...155560...547197...1736603...5150299..14554327

%C .0..1047..24490..133067...521862..1736603...5219491..14672850..39446188

%C .0..3722..86010..440282..1635481..5150299..14672850..39260363.100719254

%C .0.12816.288426.1387898..4878157.14554327..39446188.100719254.247768745

%C .0.43077.933792.4213683.14014639.39708623.102668319.251288174.594591866

%H R. H. Hardin, <a href="/A280810/b280810.txt">Table of n, a(n) for n = 1..264</a>

%F Empirical for column k:

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

%F k=2: [order 18]

%F k=3: [order 18] for n>24

%F k=4: [order 28] for n>36

%F k=5: [order 37] for n>46

%F k=6: [order 43] for n>53

%F k=7: [order 49] for n>60

%e Some solutions for n=4 k=4

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

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

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

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

%K nonn,tabl

%O 1,8

%A _R. H. Hardin_, Jan 08 2017