%I #8 Aug 10 2018 17:34:10
%S 0,3,16,92,536,3125,18216,106174,618832,3606823,21022112,122525856,
%T 714133032,4162272345,24259501048,141394733954,824108902688,
%U 4803258682187,27995443190448,163169400460516,951020959572664
%N Number of n X 3 arrays of occupancy after each element moves to some horizontal or vertical neighbor, without 2-loops or left turns.
%C Column 3 of A221768.
%H R. H. Hardin, <a href="/A221764/b221764.txt">Table of n, a(n) for n = 1..71</a>
%F Empirical: a(n) = 8*a(n-1) - 14*a(n-2) + 8*a(n-3) - a(n-4).
%F Conjectures from _Colin Barker_, Aug 10 2018: (Start)
%F G.f.: x^2*(3 - 8*x + 6*x^2) / ((1 - x)^2*(1 - 6*x + x^2)).
%F a(n) = (40+sqrt(2)*((81-56*sqrt(2))*(3+2*sqrt(2))^n - (3-2*sqrt(2))^n*(81+56*sqrt(2))) - 8*(1+n)) / 32.
%F (End)
%e Some solutions for n=3:
%e ..1..1..1....0..1..1....1..2..1....1..2..1....0..1..1....1..1..1....0..1..1
%e ..2..0..1....1..2..2....1..2..1....1..1..1....1..1..2....2..1..1....1..1..1
%e ..1..1..1....0..1..1....0..1..0....1..1..0....1..1..1....1..1..0....1..2..1
%Y Cf. A221768.
%K nonn
%O 1,2
%A _R. H. Hardin_, Jan 25 2013
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