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Number of n X 2 0..7 arrays with no element x(i,j) adjacent to itself or value 7-x(i,j) horizontally, antidiagonally or vertically, top left element zero, and 1 appearing before 2 3 4 5 and 6, 2 appearing before 3 4 and 5, and 3 appearing before 4 in row major order (unlabeled 8-colorings with no clashing color pairs).
1

%I #10 Aug 24 2024 19:46:03

%S 1,3,36,528,8256,131328,2098176,33558528,536887296,8590000128,

%T 137439215616,2199024304128,35184376283136,562949970198528,

%U 9007199321849856,144115188344291328,2305843010287435776

%N Number of n X 2 0..7 arrays with no element x(i,j) adjacent to itself or value 7-x(i,j) horizontally, antidiagonally or vertically, top left element zero, and 1 appearing before 2 3 4 5 and 6, 2 appearing before 3 4 and 5, and 3 appearing before 4 in row major order (unlabeled 8-colorings with no clashing color pairs).

%H R. H. Hardin, <a href="/A233196/b233196.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 20*a(n-1) - 64*a(n-2) for n>3.

%F Conjectures from _Colin Barker_, Oct 10 2018: (Start)

%F G.f.: x*(1 - 17*x + 40*x^2) / ((1 - 4*x)*(1 - 16*x)).

%F a(n) = 2^(2*n-7) * (4^n+8) for n>1.

%F (End)

%e Some solutions for n=5:

%e ..0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1....0..1

%e ..2..3....2..3....2..0....2..3....2..0....2..7....2..0....2..7....2..3....2..0

%e ..7..1....0..2....3..6....1..0....3..6....6..3....3..1....6..2....7..6....3..6

%e ..4..0....3..7....2..4....3..1....0..2....5..7....2..4....3..1....3..5....7..5

%e ..6..4....1..4....1..0....5..3....3..7....3..1....0..2....0..5....7..4....3..0

%Y Column 2 of A233202.

%K nonn

%O 1,2

%A _R. H. Hardin_, Dec 05 2013