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A223499 Petersen graph (3,1) coloring a rectangular array: number of n X 3 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0. 1

%I #8 Aug 21 2018 05:54:40

%S 9,115,1519,20115,266419,3528715,46737819,619042315,8199214219,

%T 108598575915,1438387920619,19051445129515,252336352607019,

%U 3342194485203115,44267359266773419,586321084882796715

%N Petersen graph (3,1) coloring a rectangular array: number of n X 3 0..5 arrays where 0..5 label nodes of a graph with edges 0,1 0,3 3,5 3,4 1,2 1,4 4,5 2,0 2,5 and every array movement to a horizontal, diagonal or antidiagonal neighbor moves along an edge of this graph, with the array starting at 0.

%C Column 3 of A223504.

%H R. H. Hardin, <a href="/A223499/b223499.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 15*a(n-1) - 24*a(n-2) + 10*a(n-3).

%F Conjectures from _Colin Barker_, Aug 21 2018: (Start)

%F G.f.: x*(9 - 20*x + 10*x^2) / ((1 - x)*(1 - 14*x + 10*x^2)).

%F a(n) = (13 + (13-2*sqrt(39))*(7-sqrt(39))^n + (7+sqrt(39))^n*(13+2*sqrt(39))) / 39.

%F (End)

%e Some solutions for n=3:

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

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

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

%Y Cf. A223504.

%K nonn

%O 1,1

%A _R. H. Hardin_, Mar 21 2013

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