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A208001 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).

%I A208001 #11 Feb 16 2025 08:33:16
%S A208001 1,2,2,5,15,5,15,114,114,15,52,1657,4141,1657,52,203,36401,426422,
%T A208001 426422,36401,203,877,1094076,86545486,450288795,86545486,1094076,877,
%U A208001 4140,42436913,29169661126,1182700979380,1182700979380,29169661126,42436913,4140
%N A208001 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 A208001 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 A208001 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KnightGraph.html">Knight Graph</a>.
%H A208001 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexColoring.html">Vertex Coloring</a>.
%e A208001 Table starts
%e A208001 ....1........2...........5............15............52.........203
%e A208001 ....2.......15.........114..........1657.........36401.....1094076
%e A208001 ....5......114........4141........426422......86545486.29169661126
%e A208001 ...15.....1657......426422.....450288795.1182700979380
%e A208001 ...52....36401....86545486.1182700979380
%e A208001 ..203..1094076.29169661126
%e A208001 ..877.42436913
%e A208001 .4140
%e A208001 ...
%e A208001 Some solutions for n=4 k=3
%e A208001 ..0..0..0....0..0..0....0..0..0....0..1..0....0..0..1....0..0..0....0..0..0
%e A208001 ..1..0..1....1..2..1....1..0..1....1..0..1....0..0..1....1..0..1....1..0..1
%e A208001 ..2..1..2....2..1..2....2..1..2....0..1..2....2..2..1....2..1..2....2..2..2
%e A208001 ..1..0..3....3..0..0....1..0..1....1..2..1....2..3..1....1..2..1....1..0..1
%Y A208001 Columns 1..4 are A000110, A207998, A207999, A208000.
%Y A208001 Main diagonal is A361453.
%Y A208001 Cf. A208434 (3 colorings), A208353 (4 colorings).
%Y A208001 Cf. A207868 (grid graph).
%K A208001 nonn,tabl
%O A208001 1,2
%A A208001 _R. H. Hardin_, Feb 22 2012