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R. H. Hardin, <a href="/A241078/b241078.txt">Table of n, a(n) for n = 1..112</a>

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T(n,k)=Number of nXk 0..2 arrays with no element equal to exactly one horizontal or vertical neighbor, with new values 0..2 introduced in row major order

1, 1, 1, 2, 4, 2, 4, 12, 12, 4, 8, 39, 67, 39, 8, 16, 131, 386, 386, 131, 16, 32, 444, 2333, 4191, 2333, 444, 32, 64, 1516, 14438, 47974, 47974, 14438, 1516, 64, 128, 5195, 90729, 572023, 1086791, 572023, 90729, 5195, 128, 256, 17847, 576616, 6996566, 25619209

1,4

Table starts

...1.....1........2...........4.............8..............16..............32

...1.....4.......12..........39...........131.............444............1516

...2....12.......67.........386..........2333...........14438...........90729

...4....39......386........4191.........47974..........572023.........6996566

...8...131.....2333.......47974.......1086791........25619209.......619056332

..16...444....14438......572023......25619209......1198954342.....57218949061

..32..1516....90729.....6996566.....619056332.....57218949061...5383469756516

..64..5195...576616....86779948...15142087661...2755743447572.510116487796210

.128.17847..3692697..1085281623..372666202588.133330360329762

.256.61424.23768052.13635536887.9201018090832

Empirical for column k:

k=1: a(n) = 2*a(n-1) for n>2

k=2: a(n) = 6*a(n-1) -8*a(n-2) -4*a(n-3) +a(n-4) +14*a(n-5) -8*a(n-6) for n>7

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

k=4: [order 96] for n>97

Some solutions for n=4 k=4

..0..0..0..1....0..1..0..2....0..1..1..0....0..1..2..2....0..1..0..1

..0..0..0..0....2..0..2..0....1..1..1..2....1..0..2..2....2..0..1..0

..0..2..1..0....1..2..1..1....1..1..1..1....0..2..1..0....1..2..0..2

..0..0..0..0....2..0..1..1....0..1..1..1....2..1..2..1....2..1..2..1

Column 1 is A000079(n-2)

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nonn,tabl

R. H. Hardin, Apr 15 2014

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