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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 knight-move neighbor (colorings ignoring permutations of colors).
6

%I #10 Mar 13 2023 17:16:05

%S 1,2,2,5,15,5,15,114,114,15,52,1657,4141,1657,52,203,36401,426422,

%T 426422,36401,203,877,1094076,86545486,450288795,86545486,1094076,877,

%U 4140,42436913,29169661126,1182700979380,1182700979380,29169661126,42436913,4140

%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 knight-move neighbor (colorings ignoring permutations of colors).

%H Andrew Howroyd, <a href="/A208001/b208001.txt">Table of n, a(n) for n = 1..66</a> (terms 1..39 from R. H. Hardin).

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KnightGraph.html">Knight Graph</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexColoring.html">Vertex Coloring</a>.

%e Table starts

%e ....1........2...........5............15............52.........203

%e ....2.......15.........114..........1657.........36401.....1094076

%e ....5......114........4141........426422......86545486.29169661126

%e ...15.....1657......426422.....450288795.1182700979380

%e ...52....36401....86545486.1182700979380

%e ..203..1094076.29169661126

%e ..877.42436913

%e .4140

%e ...

%e Some solutions for n=4 k=3

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

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

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

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

%Y Columns 1..4 are A000110, A207998, A207999, A208000.

%Y Main diagonal is A361453.

%Y Cf. A208434 (3 colorings), A208353 (4 colorings).

%Y Cf. A207868 (grid graph).

%K nonn,tabl

%O 1,2

%A _R. H. Hardin_, Feb 22 2012