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A223352 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.

%I A223352 #6 Jul 23 2025 04:02:24
%S A223352 6,12,12,28,52,28,60,236,236,60,140,1076,2280,1076,140,300,4908,20836,
%T A223352 20836,4908,300,700,22388,202264,405988,202264,22388,700,1500,102124,
%U A223352 1851020,7918948,7918948,1851020,102124,1500,3500,465844,17970056
%N A223352 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 A223352 Table starts
%C A223352 ....6......12..........28.............60...............140..................300
%C A223352 ...12......52.........236...........1076..............4908................22388
%C A223352 ...28.....236........2280..........20836............202264..............1851020
%C A223352 ...60....1076.......20836.........405988...........7918948............154482340
%C A223352 ..140....4908......202264........7918948.........329268616..........12912752876
%C A223352 ..300...22388.....1851020......154482340.......12912752876........1079538816324
%C A223352 ..700..102124....17970056.....3013692516......537001573656.......90254934876620
%C A223352 .1500..465844...164457412....58792282660....21060038730884.....7545802995190884
%C A223352 .3500.2124972..1596586328..1146943179236...875825392204488...630870570544801836
%C A223352 .7500.9693172.14611562156.22375024222628.34348003384801484.52744249735952687492
%H A223352 R. H. Hardin, <a href="/A223352/b223352.txt">Table of n, a(n) for n = 1..480</a>
%F A223352 Empirical for column k:
%F A223352 k=1: a(n) = 5*a(n-2) for n>3
%F A223352 k=2: a(n) = 5*a(n-1) -2*a(n-2)
%F A223352 k=3: a(n) = 91*a(n-2) -192*a(n-4) +64*a(n-6)
%F A223352 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 A223352 k=5: [order 12] for n>13
%F A223352 k=6: [order 18]
%F A223352 k=7: [order 36]
%e A223352 Some solutions for n=3 k=4
%e A223352 ..1..0..1..4....1..0..1..4....2..0..1..0....3..1..0..1....4..1..0..2
%e A223352 ..4..2..4..2....3..1..0..2....4..1..0..2....1..3..1..0....1..0..2..4
%e A223352 ..1..0..1..4....1..4..2..0....2..4..2..4....0..1..0..2....4..2..4..1
%Y A223352 Column 2 is A223249
%K A223352 nonn,tabl
%O A223352 1,1
%A A223352 _R. H. Hardin_ Mar 19 2013