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A223291 4-loop graph coloring a rectangular array: number of n X 2 0..8 arrays where 0..8 label nodes of a graph with edges 0,1 1,2 2,0 0,3 3,4 4,0 0,5 5,6 6,0 0,7 7,8 8,0 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph. 1

%I

%S 24,168,1368,11304,93528,773928,6404184,52994088,438521688,3628730664,

%T 30027445848,248474628648,2056106982744,17014115072808,

%U 140790393758808,1165028853392424,9640517317978968,79774482741456168,660127240765231704

%N 4-loop graph coloring a rectangular array: number of n X 2 0..8 arrays where 0..8 label nodes of a graph with edges 0,1 1,2 2,0 0,3 3,4 4,0 0,5 5,6 6,0 0,7 7,8 8,0 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

%C Column 2 of A223297.

%H R. H. Hardin, <a href="/A223291/b223291.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 9*a(n-1) - 6*a(n-2).

%F Conjectures from _Colin Barker_, Aug 19 2018: (Start)

%F G.f.: 24*x*(1 - 2*x) / (1 - 9*x + 6*x^2).

%F a(n) = (2^(2-n)*((9-sqrt(57))^n*(3+sqrt(57)) + (-3+sqrt(57))*(9+sqrt(57))^n)) / sqrt(57).

%F (End)

%e Some solutions for n=3:

%e ..2..0....0..3....6..0....4..0....3..0....0..3....0..1....4..0....8..0....0..2

%e ..0..1....3..0....0..3....0..4....0..5....6..0....8..0....0..6....0..3....2..0

%e ..7..0....0..6....8..0....4..3....1..0....0..6....7..8....3..0....8..0....0..6

%Y Cf. A223297.

%K nonn

%O 1,1

%A _R. H. Hardin_, Mar 19 2013

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Last modified September 26 14:21 EDT 2022. Contains 356999 sequences. (Running on oeis4.)