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Number of n X 2 arrays of occupancy after each element stays put or moves to some horizontal, vertical or antidiagonal neighbor, with no occupancy greater than 2.
1

%I #9 Aug 05 2018 08:21:28

%S 3,19,139,1035,7723,57643,430251,3211435,23970475,178918059,

%T 1335462571,9968028331,74402376363,555346897579,4145165675179,

%U 30939937811115,230938839788203,1723750967061163,12866252377336491,96035015150447275

%N Number of n X 2 arrays of occupancy after each element stays put or moves to some horizontal, vertical or antidiagonal neighbor, with no occupancy greater than 2.

%C Column 2 of A221378.

%H R. H. Hardin, <a href="/A221374/b221374.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 9*a(n-1) - 12*a(n-2) + 4*a(n-3).

%F Conjectures from _Colin Barker_, Aug 05 2018: (Start)

%F G.f.: x*(3 - 2*x)*(1 - 2*x) / ((1 - x)*(1 - 8*x + 4*x^2)).

%F a(n) = (1 + (4-2*sqrt(3))^n + (2*(2+sqrt(3)))^n) / 3.

%F (End)

%e Some solutions for n=3:

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

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

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

%Y Cf. A221378.

%K nonn

%O 1,1

%A _R. H. Hardin_, Jan 13 2013