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%I #17 May 10 2018 11:02:20
%S 1,1,1,2,3,2,4,9,9,4,8,27,41,27,8,16,81,187,187,81,16,32,243,853,1302,
%T 853,243,32,64,729,3891,9075,9075,3891,729,64,128,2187,17749,63267,
%U 96831,63267,17749,2187,128,256,6561,80963,441090,1034073,1034073,441090,80963
%N T(n,k) = number of n X k 0..2 arrays with new values 0..2 introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).
%C Number of colorings of the grid graph P_n X P_k using a maximum of 3 colors up to permutation of the colors. - _Andrew Howroyd_, Jun 26 2017
%H R. H. Hardin, <a href="/A207997/b207997.txt">Table of n, a(n) for n = 1..544</a>
%H R. J. Mathar, <a href="http://vixra.org/abs/1511.0225">Counting 2-way monotonic terrace forms over rectangular landscapes</a>, vixra:1511.0225, eq. (33)-(35).
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VertexColoring.html">Vertex Coloring</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Graph_coloring">Graph Coloring</a>
%F 2*T(n,m) = A078099(n,m) for m>1. - _R. J. Mathar_, Nov 23 2015
%e Table starts
%e ..1....1.....2.......4.........8.........16...........32............64
%e ..1....3.....9......27........81........243..........729..........2187
%e ..2....9....41.....187.......853.......3891........17749.........80963
%e ..4...27...187....1302......9075......63267.......441090.......3075255
%e ..8...81...853....9075.....96831....1034073.....11045757.....117997043
%e .16..243..3891...63267...1034073...16932816....277458045....4547477370
%e .32..729.17749..441090..11045757..277458045...6978332618..175605187731
%e .64.2187.80963.3075255.117997043.4547477370.175605187731.6787438272198
%e ...
%e Some solutions for n=4, k=3:
%e ..0..1..2....0..1..0....0..1..0....0..1..2....0..1..2....0..1..2....0..1..0
%e ..2..0..1....2..0..2....1..0..2....1..2..1....2..0..1....1..2..1....1..2..1
%e ..0..2..0....0..1..0....2..1..0....0..1..2....0..2..0....0..1..2....2..0..2
%e ..1..0..1....1..2..1....1..0..1....1..2..0....2..0..2....2..0..1....1..2..0
%Y Cf. A020698 (column 3), A078100 (column 4), A207994 (column 5), A207995 (column 6), A207996 (column 7).
%Y Main diagonal is A207993.
%Y Cf. A198715 (4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings), A207868 (unlimited).
%K nonn,tabl
%O 1,4
%A _R. H. Hardin_, Feb 22 2012
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