%I #15 Feb 16 2025 08:33:16

%S 1,1,1,2,4,2,5,25,25,5,14,172,401,172,14,41,1201,6548,6548,1201,41,

%T 122,8404,107042,250031,107042,8404,122,365,58825,1749965,9548295,

%U 9548295,1749965,58825,365,1094,411772,28609241,364637102,851787199,364637102

%N T(n,k)=Number of nXk 0..3 arrays with values 0..3 introduced in row major order and no element equal to any horizontal or vertical neighbor.

%C Number of colorings of the grid graph P_n X P_k using a maximum of 4 colors up to permutation of the colors. - _Andrew Howroyd_, Jun 26 2017

%H Andrew Howroyd, <a href="/A198715/b198715.txt">Table of n, a(n) for n = 1..496</a> (terms 1..180 from R. H. Hardin)

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

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

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Graph_coloring">Graph Coloring</a>

%e Table starts

%e ....1........1............2...............5..................14

%e ....1........4...........25.............172................1201

%e ....2.......25..........401............6548..............107042

%e ....5......172.........6548..........250031.............9548295

%e ...14.....1201.......107042.........9548295...........851787199

%e ...41.....8404......1749965.......364637102.........75987485516

%e ..122....58825.....28609241.....13925032958.......6778819400772

%e ..365...411772....467717288....531779578441.....604736581320925

%e .1094..2882401...7646461682..20307996787865...53948385378521909

%e .3281.20176804.125007943505.775536991678112.4812720805166620356

%e ...

%e Some solutions with all values from 0 to 3 for n=6 k=4

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

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

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

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

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

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

%Y Columns 1-7 are A007051(n-2), A034494(n-1), A198710, A198711, A198712, A198713, A198714.

%Y Main diagonal is A198709.

%Y Cf. A207997 (3 colorings), A222444 (labeled 4 colorings), A198906 (5 colorings), A198982 (6 colorings), A198723 (7 colorings), A198914 (8 colorings), A207868 (unlimited).

%K nonn,tabl,changed

%O 1,4

%A _R. H. Hardin_, Oct 29 2011