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T(n,k) = Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).
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%I #11 Jun 25 2017 17:22:51

%S 1,1,1,2,2,2,5,15,15,5,15,203,716,203,15,52,4140,83440,83440,4140,52,

%T 203,115975,18171918,112073062,18171918,115975,203,877,4213597,

%U 6423127757,346212384169,346212384169,6423127757,4213597,877,4140,190899322

%N T(n,k) = Number of n X k nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal, vertical or antidiagonal neighbor (colorings ignoring permutations of colors).

%C Equivalently, the number of colorings in the rhombic hexagonal square grid graph RH_(n,k) using any number of colors up to permutation of the colors. - _Andrew Howroyd_, Jun 25 2017

%H Andrew Howroyd, <a href="/A208054/b208054.txt">Table of n, a(n) for n = 1..231</a> (terms 1..49 from R. H. Hardin)

%e Table starts

%e ...1.........1.............2................5................15

%e ...1.........2............15..............203..............4140

%e ...2........15...........716............83440..........18171918

%e ...5.......203.........83440........112073062......346212384169

%e ..15......4140......18171918.....346212384169.18633407199331522

%e ..52....115975....6423127757.2043836452962923

%e .203...4213597.3376465219485

%e .877.190899322

%e ...

%e Some solutions for n=4 k=3

%e ..0..1..0....0..1..0....0..1..0....0..1..0....0..1..2....0..1..0....0..1..0

%e ..2..3..1....2..3..4....2..3..2....2..3..1....2..3..0....2..3..1....2..3..2

%e ..4..2..4....0..5..0....0..4..0....0..4..5....4..5..3....4..5..3....0..1..4

%e ..0..5..0....1..2..1....1..2..1....5..3..4....0..1..0....0..6..4....2..0..1

%Y Columns 1-5 are A000110(n-1), A020557(n-1), A208051, A208052, A208053.

%Y Cf. A207868, A212162, A212163, A208050, A068271.

%K nonn,tabl

%O 1,4

%A _R. H. Hardin_, Feb 22 2012