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T(n,k)=3X3X3 triangular graph without horizontal edges coloring a rectangular array: number of nXk 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph
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%I #4 Mar 19 2013 13:06:11

%S 6,12,12,28,52,28,60,236,236,60,140,1076,2280,1076,140,300,4908,20836,

%T 20836,4908,300,700,22388,202264,405988,202264,22388,700,1500,102124,

%U 1851020,7918948,7918948,1851020,102124,1500,3500,465844,17970056

%N T(n,k)=3X3X3 triangular graph without horizontal edges coloring a rectangular array: number of nXk 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,2 1,3 1,4 2,4 2,5 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph

%C Table starts

%C ....6......12..........28.............60...............140..................300

%C ...12......52.........236...........1076..............4908................22388

%C ...28.....236........2280..........20836............202264..............1851020

%C ...60....1076.......20836.........405988...........7918948............154482340

%C ..140....4908......202264........7918948.........329268616..........12912752876

%C ..300...22388.....1851020......154482340.......12912752876........1079538816324

%C ..700..102124....17970056.....3013692516......537001573656.......90254934876620

%C .1500..465844...164457412....58792282660....21060038730884.....7545802995190884

%C .3500.2124972..1596586328..1146943179236...875825392204488...630870570544801836

%C .7500.9693172.14611562156.22375024222628.34348003384801484.52744249735952687492

%H R. H. Hardin, <a href="/A223352/b223352.txt">Table of n, a(n) for n = 1..480</a>

%F Empirical for column k:

%F k=1: a(n) = 5*a(n-2) for n>3

%F k=2: a(n) = 5*a(n-1) -2*a(n-2)

%F k=3: a(n) = 91*a(n-2) -192*a(n-4) +64*a(n-6)

%F k=4: a(n) = 23*a(n-1) -66*a(n-2) -52*a(n-3) +208*a(n-4) +32*a(n-5) -128*a(n-6)

%F k=5: [order 12] for n>13

%F k=6: [order 18]

%F k=7: [order 36]

%e Some solutions for n=3 k=4

%e ..1..0..1..4....1..0..1..4....2..0..1..0....3..1..0..1....4..1..0..2

%e ..4..2..4..2....3..1..0..2....4..1..0..2....1..3..1..0....1..0..2..4

%e ..1..0..1..4....1..4..2..0....2..4..2..4....0..1..0..2....4..2..4..1

%Y Column 2 is A223249

%K nonn,tabl

%O 1,1

%A _R. H. Hardin_ Mar 19 2013