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A208021 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, diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).

%I A208021 #14 Feb 16 2025 08:33:16
%S A208021 1,1,1,2,1,2,5,7,7,5,15,87,270,87,15,52,1657,27093,27093,1657,52,203,
%T A208021 43833,5252041,30066912,5252041,43833,203,877,1515903,1688298227,
%U A208021 80318704605,80318704605,1688298227,1515903,877,4140,65766991
%N A208021 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, diagonal or antidiagonal neighbor (colorings ignoring permutations of colors).
%C A208021 Equivalently, the number of colorings of the n x k king graph using any number of colors up to permutation of the colors. - _Andrew Howroyd_, Jun 25 2017
%H A208021 Andrew Howroyd, <a href="/A208021/b208021.txt">Table of n, a(n) for n = 1..231</a> (terms 1..49 from R. H. Hardin)
%H A208021 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/KingGraph.html">King Graph</a>
%e A208021 Table starts
%e A208021 ...1........1............2...............5...............15..............52
%e A208021 ...1........1............7..............87.............1657...........43833
%e A208021 ...2........7..........270...........27093..........5252041......1688298227
%e A208021 ...5.......87........27093........30066912......80318704605.421673189900658
%e A208021 ..15.....1657......5252041.....80318704605.3662498214110836
%e A208021 ..52....43833...1688298227.421673189900658
%e A208021 .203..1515903.819147302097
%e A208021 .877.65766991
%e A208021 ...
%e A208021 Some solutions for n=4 k=3
%e A208021 ..0..1..2....0..1..0....0..1..0....0..1..0....0..1..0....0..1..0....0..1..2
%e A208021 ..2..3..4....2..3..2....2..3..2....2..3..2....2..3..2....2..3..2....2..3..0
%e A208021 ..5..6..0....4..0..4....0..1..0....4..1..0....0..1..0....0..4..0....4..5..4
%e A208021 ..2..3..1....1..2..1....2..3..4....5..2..3....2..4..2....1..2..1....0..1..2
%Y A208021 Columns 1-5 are A000110(n-1), A020556(n-1), A208018, A208019, A208020.
%Y A208021 Main diagonal is A289136.
%Y A208021 Cf. A207868, A212208, A212209.
%K A208021 nonn,tabl
%O A208021 1,4
%A A208021 _R. H. Hardin_, Feb 22 2012