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A223249 Two-loop graph coloring a rectangular array: number of n X 2 0..4 arrays where 0..4 label nodes of a graph with edges 0,1 1,2 2,0 0,3 3,4 4,0 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.

%I A223249 #8 Mar 16 2018 06:55:58
%S A223249 12,52,236,1076,4908,22388,102124,465844,2124972,9693172,44215916,
%T A223249 201693236,920034348,4196785268,19143857644,87325717684,398340873132,
%U A223249 1817052930292,8288582905196,37808808665396,172466877516588
%N A223249 Two-loop graph coloring a rectangular array: number of n X 2 0..4 arrays where 0..4 label nodes of a graph with edges 0,1 1,2 2,0 0,3 3,4 4,0 and every array movement to a horizontal or vertical neighbor moves along an edge of this graph.
%C A223249 Column 2 of A223255.
%H A223249 R. H. Hardin, <a href="/A223249/b223249.txt">Table of n, a(n) for n = 1..210</a>
%F A223249 Empirical: a(n) = 5*a(n-1) - 2*a(n-2).
%F A223249 Conjectures from _Colin Barker_, Mar 16 2018: (Start)
%F A223249 G.f.: 4*x*(3 - 2*x) / (1 - 5*x + 2*x^2).
%F A223249 a(n) = (2^(1-n)*((5-sqrt(17))^n*(-1+sqrt(17)) + (1+sqrt(17))*(5+sqrt(17))^n)) / sqrt(17).
%F A223249 (End)
%e A223249 Some solutions for n=3:
%e A223249 ..3..4....4..0....1..0....0..2....4..0....2..0....0..3....4..0....3..4....1..2
%e A223249 ..4..0....0..2....0..4....3..0....0..2....0..2....2..0....0..4....4..0....0..1
%e A223249 ..0..3....4..0....4..3....4..3....1..0....4..0....0..4....2..0....0..2....1..0
%Y A223249 Cf. A223255.
%K A223249 nonn
%O A223249 1,1
%A A223249 _R. H. Hardin_, Mar 18 2013