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A223556 T(n,k)=Petersen graph (3,1) coloring a rectangular array: number of nXk 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0.

%I A223556 #6 Jun 02 2025 08:29:50
%S A223556 1,3,6,9,27,36,27,171,243,216,81,1089,3249,2187,1296,243,6939,44217,
%T A223556 61731,19683,7776,729,44217,609309,1795473,1172889,177147,46656,2187,
%U A223556 281763,8410671,53599905,72906921,22284891,1594323,279936,6561,1795473
%N A223556 T(n,k)=Petersen graph (3,1) coloring a rectangular array: number of nXk 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0.
%C A223556 Table starts
%C A223556 ........1..........3.............9...............27...................81
%C A223556 ........6.........27...........171.............1089.................6939
%C A223556 .......36........243..........3249............44217...............609309
%C A223556 ......216.......2187.........61731..........1795473.............53599905
%C A223556 .....1296......19683.......1172889.........72906921...........4715559621
%C A223556 .....7776.....177147......22284891.......2960456193.........414863325945
%C A223556 ....46656....1594323.....423412929.....120212193177.......36498667573629
%C A223556 ...279936...14348907....8044845651....4881332621169.....3211064180380305
%C A223556 ..1679616..129140163..152852067369..198211242377097...282501632829717621
%C A223556 .10077696.1162261467.2904189280011.8048559615522273.24853807982558115945
%H A223556 R. H. Hardin, <a href="/A223556/b223556.txt">Table of n, a(n) for n = 1..219</a>
%F A223556 Empirical for column k:
%F A223556 k=1: a(n) = 6*a(n-1)
%F A223556 k=2: a(n) = 9*a(n-1)
%F A223556 k=3: a(n) = 19*a(n-1)
%F A223556 k=4: a(n) = 41*a(n-1) -16*a(n-2)
%F A223556 k=5: a(n) = 95*a(n-1) -626*a(n-2) +720*a(n-3) for n>4
%F A223556 k=6: [order 8] for n>9
%F A223556 k=7: [order 13] for n>15
%F A223556 Empirical for row n:
%F A223556 n=1: a(n) = 3*a(n-1)
%F A223556 n=2: a(n) = 7*a(n-1) -4*a(n-2) for n>3
%F A223556 n=3: a(n) = 17*a(n-1) -47*a(n-2) +41*a(n-3) -10*a(n-4) for n>6
%F A223556 n=4: [order 13] for n>16
%F A223556 n=5: [order 41] for n>45
%e A223556 Some solutions for n=3 k=4
%e A223556 ..0..3..5..2....0..1..4..1....0..2..5..2....0..2..5..2....0..1..2..0
%e A223556 ..5..3..0..1....0..1..0..3....1..2..5..2....5..2..1..4....2..0..1..2
%e A223556 ..4..3..4..3....2..1..4..5....0..2..0..1....1..2..1..2....1..4..5..3
%Y A223556 Column 1 is A000400(n-1)
%Y A223556 Column 2 is A013708(n-1)
%Y A223556 Column 3 = 9*19^(n-1) is row 8 of A223556 with T(2+,3) = A121057(8,1+)
%Y A223556 Row 1 is A000244(n-1)
%K A223556 nonn,tabl
%O A223556 1,2
%A A223556 _R. H. Hardin_ Mar 22 2013