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Number of n X 3 0..5 arrays with no element x(i,j) adjacent to value 5-x(i,j) horizontally or antidiagonally, top left element zero, and 1 appearing before 2 3 and 4, and 2 appearing before 3 in row major order.
1

%I #7 Oct 10 2018 10:01:01

%S 6,376,35446,3638416,380283286,39892988056,4187991850726,

%T 439722934529296,46170567719841766,4847902462062966136,

%U 509029608397682640406,53448105729259172362576,5612051035369765495590646

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

%H R. H. Hardin, <a href="/A233212/b233212.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 127*a(n-1) - 2331*a(n-2) + 2205*a(n-3).

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

%F G.f.: 2*x*(3 - 193*x + 840*x^2) / ((1 - x)*(1 - 21*x)*(1 - 105*x)).

%F a(n) = (105 + 2*3^(2+n)*7^n + 5^(1+n)*21^n) / 168.

%F (End)

%e Some solutions for n=3:

%e ..0..1..5....0..1..0....0..1..5....0..0..1....0..1..2....0..1..2....0..0..1

%e ..5..2..2....2..2..0....1..2..1....1..2..1....5..1..2....2..2..2....2..0..1

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

%Y Column 3 of A233217.

%K nonn

%O 1,1

%A _R. H. Hardin_, Dec 06 2013