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A223504 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, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0.

%I A223504 #6 Jun 02 2025 08:28:46
%S A223504 1,3,6,9,19,36,27,115,121,216,81,631,1519,771,1296,243,3539,16323,
%T A223504 20115,4913,7776,729,19759,182901,426359,266419,31307,46656,2187,
%U A223504 110427,2030665,9685063,11148439,3528715,199497,279936,6561,617015,22598167
%N A223504 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, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0.
%C A223504 Table starts
%C A223504 ........1........3............9..............27.................81
%C A223504 ........6.......19..........115.............631...............3539
%C A223504 .......36......121.........1519...........16323.............182901
%C A223504 ......216......771........20115..........426359............9685063
%C A223504 .....1296.....4913.......266419........11148439..........515473927
%C A223504 .....7776....31307......3528715.......291545903........27465794119
%C A223504 ....46656...199497.....46737819......7624417031......1463848507173
%C A223504 ...279936..1271251....619042315....199391762123.....78024299447333
%C A223504 ..1679616..8100769...8199214219...5214442630935...4158831849750231
%C A223504 .10077696.51620379.108598575915.136366781617267.221674060909378867
%H A223504 R. H. Hardin, <a href="/A223504/b223504.txt">Table of n, a(n) for n = 1..199</a>
%F A223504 Empirical for column k:
%F A223504 k=1: a(n) = 6*a(n-1)
%F A223504 k=2: a(n) = 7*a(n-1) -4*a(n-2)
%F A223504 k=3: a(n) = 15*a(n-1) -24*a(n-2) +10*a(n-3)
%F A223504 k=4: a(n) = 31*a(n-1) -127*a(n-2) -20*a(n-3) +705*a(n-4) -1027*a(n-5) +499*a(n-6) -60*a(n-7)
%F A223504 k=5: [order 21]
%F A223504 k=6: [order 53]
%F A223504 Empirical for row n:
%F A223504 n=1: a(n) = 3*a(n-1)
%F A223504 n=2: a(n) = 5*a(n-1) +4*a(n-2) -4*a(n-3) for n>4
%F A223504 n=3: a(n) = 12*a(n-1) -4*a(n-2) -73*a(n-3) +103*a(n-4) -23*a(n-5) -16*a(n-6) +4*a(n-7) for n>8
%F A223504 n=4: [order 21] for n>22
%F A223504 n=5: [order 60] for n>61
%e A223504 Some solutions for n=3 k=4
%e A223504 ..0..3..4..1....0..2..1..4....0..3..0..3....0..2..1..2....0..1..4..3
%e A223504 ..0..3..4..3....5..2..5..4....4..1..0..1....1..2..0..2....0..1..0..3
%e A223504 ..5..3..0..1....1..2..1..2....0..1..0..1....5..2..0..2....0..3..0..1
%Y A223504 Column 1 is A000400(n-1)
%Y A223504 Column 2 is A138977
%Y A223504 Row 1 is A000244(n-1)
%K A223504 nonn,tabl
%O A223504 1,2
%A A223504 _R. H. Hardin_ Mar 21 2013