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6 X 6 X 6 triangular graph coloring a rectangular array: number of n X 1 0..20 arrays where 0..20 label nodes of the fully triangulated graph and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.
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%I #8 Aug 20 2018 05:31:03

%S 21,90,420,1992,9552,45984,221760,1070208,5166336,24943104,120431616,

%T 581486592,2807648256,13556490240,65456455680,316051587072,

%U 1526031777792,7368332673024,35577456230400,171783152467968,829442428502016

%N 6 X 6 X 6 triangular graph coloring a rectangular array: number of n X 1 0..20 arrays where 0..20 label nodes of the fully triangulated graph and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

%C Column 1 of A223370.

%H R. H. Hardin, <a href="/A223363/b223363.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 6*a(n-1) - 4*a(n-2) - 8*a(n-3).

%F Conjectures from _Colin Barker_, Aug 20 2018: (Start)

%F G.f.: 3*x*(7 - 12*x - 12*x^2) / ((1 - 2*x)*(1 - 4*x - 4*x^2)).

%F a(n) = (3/2)*(2^n-(2-2*sqrt(2))^n*(-1+sqrt(2)) + 2^(1/2+n)*(1+sqrt(2))^n + (2*(1+sqrt(2)))^n).

%F (End)

%e Some solutions for n=3:

%e .20....2....4....4...18....4....6....7....6...17...14....4...11....8...15....5

%e .14....5....7....7...13....3....3....8....3...16...13....3...10....5...10....8

%e .20....2....8...11...14....1....4...12....6...11....9....6...16....2...11....5

%e Vertex neighbors:

%e 0 -> 1 2

%e 1 -> 0 2 3 4

%e 2 -> 0 1 4 5

%e 3 -> 1 4 6 7

%e 4 -> 1 2 3 5 7 8

%e 5 -> 2 4 8 9

%e 6 -> 3 7 10 11

%e 7 -> 3 4 6 8 11 12

%e 8 -> 4 5 7 9 12 13

%e 9 -> 5 8 13 14

%e 10 -> 6 11 15 16

%e 11 -> 6 7 10 12 16 17

%e 12 -> 7 8 11 13 17 18

%e 13 -> 8 9 12 14 18 19

%e 14 -> 9 13 19 20

%e 15 -> 10 16

%e 16 -> 10 11 15 17

%e 17 -> 11 12 16 18

%e 18 -> 12 13 17 19

%e 19 -> 13 14 18 20

%e 20 -> 14 19

%Y Cf. A223370.

%K nonn

%O 1,1

%A _R. H. Hardin_, Mar 19 2013