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A207868 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 or vertical neighbor (colorings ignoring permutations of colors).

%I A207868 #12 Feb 16 2025 08:33:16
%S A207868 1,1,1,2,4,2,5,34,34,5,15,500,2052,500,15,52,10900,278982,278982,
%T A207868 10900,52,203,322768,68162042,455546040,68162042,322768,203,877,
%U A207868 12297768,26419793726,1625686993918,1625686993918,26419793726,12297768,877
%N A207868 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 or vertical neighbor (colorings ignoring permutations of colors).
%C A207868 Table starts
%C A207868 ...1.........1..............2.................5.................15
%C A207868 ...1.........4.............34...............500..............10900
%C A207868 ...2........34...........2052............278982...........68162042
%C A207868 ...5.......500.........278982.........455546040......1625686993918
%C A207868 ..15.....10900.......68162042.....1625686993918.103204230192540988
%C A207868 ..52....322768....26419793726.10764437129618296
%C A207868 .203..12297768.15002771641712
%C A207868 .877.580849872
%H A207868 Andrew Howroyd, <a href="/A207868/b207868.txt">Table of n, a(n) for n = 1..231</a> (terms 1..49 from R. H. Hardin)
%H A207868 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>.
%H A207868 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexColoring.html">Vertex Coloring</a>.
%H A207868 Wikipedia, <a href="https://en.wikipedia.org/wiki/Graph_coloring">Graph Coloring</a>.
%e A207868 Some solutions for n=5 k=3
%e A207868 ..0..1..0....0..1..2....0..1..0....0..1..0....0..1..2....0..1..0....0..1..0
%e A207868 ..1..0..1....1..0..3....1..0..1....1..0..1....1..2..0....1..0..1....1..0..1
%e A207868 ..0..1..0....0..1..0....0..1..0....2..1..0....0..1..2....0..2..3....0..1..2
%e A207868 ..1..0..1....1..0..1....1..0..1....0..2..3....1..0..1....1..0..1....1..0..1
%e A207868 ..0..1..0....0..1..0....2..1..0....1..3..0....2..1..0....0..1..0....0..1..0
%Y A207868 Columns 1..5 are A000110(n-1), A207864, A207865, A207866, A207867.
%Y A207868 Main diagonal is A207863.
%Y A207868 Cf. A207997 (3 colorings), A198715 (4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings).
%Y A207868 Cf. A207981, A208001 (knight), A208021 (king), A208054, A208096, A208301.
%K A207868 nonn,tabl
%O A207868 1,4
%A A207868 _R. H. Hardin_, Feb 21 2012