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A222444 T(n,k) = number of n X k 0..3 arrays with entries increasing mod 4 by 0, 1 or 2 rightwards and downwards, starting with upper left zero.

%I A222444 #18 Feb 16 2025 08:33:19
%S A222444 1,3,3,9,21,9,27,147,147,27,81,1029,2403,1029,81,243,7203,39285,39285,
%T A222444 7203,243,729,50421,642249,1500183,642249,50421,729,2187,352947,
%U A222444 10499787,57289767,57289767,10499787,352947,2187,6561,2470629,171655443
%N A222444 T(n,k) = number of n X k 0..3 arrays with entries increasing mod 4 by 0, 1 or 2 rightwards and downwards, starting with upper left zero.
%C A222444 1/4 the number of 4-colorings of the grid graph P_n X P_k. - _Andrew Howroyd_, Jun 26 2017
%H A222444 Andrew Howroyd, <a href="/A222444/b222444.txt">Table of n, a(n) for n = 1..496</a> (terms 1..180 from R. H. Hardin)
%H A222444 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GridGraph.html">Grid Graph</a>
%H A222444 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexColoring.html">Vertex Coloring</a>
%H A222444 Wikipedia, <a href="https://en.wikipedia.org/wiki/Graph_coloring">Graph Coloring</a>
%F A222444 T(n,k) = 6*A198715(n,k) - 3 for n*k>1. - _Andrew Howroyd_, Jun 27 2017
%F A222444 Empirical for column k:
%F A222444 k=1: a(n) = 3*a(n-1).
%F A222444 k=2: a(n) = 7*a(n-1).
%F A222444 k=3: a(n) = 18*a(n-1) - 27*a(n-2).
%F A222444 k=4: a(n) = 45*a(n-1) - 267*a(n-2) + 263*a(n-3).
%F A222444 k=5: a(n) = 118*a(n-1) - 2811*a(n-2) + 22255*a(n-3) - 53860*a(n-4) - 54747*a(n-5) + 269406*a(n-6) - 175392*a(n-7).
%F A222444 k=6: [order 13]
%F A222444 k=7: [order 32]
%e A222444 Table starts
%e A222444 ......1..........3...............9..................27.......................81
%e A222444 ......3.........21.............147................1029.....................7203
%e A222444 ......9........147............2403...............39285...................642249
%e A222444 .....27.......1029...........39285.............1500183.................57289767
%e A222444 .....81.......7203..........642249............57289767...............5110723191
%e A222444 ....243......50421........10499787..........2187822609.............455924913093
%e A222444 ....729.....352947.......171655443.........83550197745...........40672916404629
%e A222444 ...2187....2470629......2806303725.......3190677470643.........3628419487925547
%e A222444 ...6561...17294403.....45878770089.....121847980727187.......323690312271131451
%e A222444 ..19683..121060821....750047661027....4653221950068669.....28876324830999722133
%e A222444 ..59049..847425747..12262131106083..177700725073710285...2576049100980154511889
%e A222444 .177147.5931980229.200467073061765.6786168386579878383.229808641254065144560647
%e A222444 ...
%e A222444 Some solutions for n=3, k=4:
%e A222444 ..0..0..0..2....0..0..2..0....0..2..0..0....0..2..0..2....0..0..2..3
%e A222444 ..1..2..2..3....0..2..3..1....2..2..2..0....0..0..0..2....0..2..3..1
%e A222444 ..2..2..3..1....2..0..1..3....2..2..0..0....2..0..1..3....1..2..0..1
%Y A222444 Columns 1-7 are A000244(n-1), A169634(n-1), A222439, A222440, A222441, A222442, A222443.
%Y A222444 Main diagonal is A068254.
%Y A222444 Cf. A078099 (3 colorings), A198715 (unlabeled 4 colorings), A222144 (5 colorings), A222281 (6 colorings), A222340 (7 colorings), A222462 (8 colorings).
%K A222444 nonn,tabl
%O A222444 1,2
%A A222444 _R. H. Hardin_, Feb 20 2013