login
Two-loop graph coloring a rectangular array: number of n X 4 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.
1

%I #8 Aug 18 2018 08:43:19

%S 80,1076,17828,307144,5359892,93770308,1641741608,28748561780,

%T 503440061060,8816254627208,154390919319636,2703707386173764,

%U 47347570829880040,829155025056272692,14520239405020681988

%N Two-loop graph coloring a rectangular array: number of n X 4 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 Column 4 of A223255.

%H R. H. Hardin, <a href="/A223251/b223251.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 20*a(n-1) - 28*a(n-2) - 299*a(n-3) + 436*a(n-4) + 476*a(n-5) - 460*a(n-6) for n>7.

%F Empirical g.f.: 4*x*(20 - 131*x - 363*x^2 + 1158*x^3 + 760*x^4 - 1036*x^5 + 24*x^6) / (1 - 20*x + 28*x^2 + 299*x^3 - 436*x^4 - 476*x^5 + 460*x^6). - _Colin Barker_, Aug 18 2018

%e Some solutions for n=3:

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

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

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

%Y Cf. A223255.

%K nonn

%O 1,1

%A _R. H. Hardin_, Mar 18 2013