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A220639 Number of ways to reciprocally link elements of an n X 3 array either to themselves or to exactly one king-move neighbor.

%I A220639 #16 Nov 03 2023 15:23:24
%S A220639 1,3,40,369,3755,37320,373177,3725843,37213728,371654153,3711809483,
%T A220639 37070598992,370232236753,3697589375491,36928628181272,
%U A220639 368814220524417,3683427651446923,36787191180049816,367401660507886793,3669320102980547411,36646296045314442000
%N A220639 Number of ways to reciprocally link elements of an n X 3 array either to themselves or to exactly one king-move neighbor.
%C A220639 Column 3 of A220644.
%H A220639 Alois P. Heinz, <a href="/A220639/b220639.txt">Table of n, a(n) for n = 0..500</a> (terms n = 1..210 from R. H. Hardin)
%H A220639 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (8,22,-20,-16,10,-3).
%F A220639 Empirical: a(n) = 8*a(n-1) +22*a(n-2) -20*a(n-3) -16*a(n-4) +10*a(n-5) -3*a(n-6).
%F A220639 G.f.: -(x^4-3*x^3+6*x^2+5*x-1)/((x-1)*(3*x^5-7*x^4+9*x^3+29*x^2+7*x-1)). - _Alois P. Heinz_, Jun 03 2014
%e A220639 Some solutions for n=3 0=self 1=nw 2=n 3=ne 4=w 6=e 7=sw 8=s 9=se (reciprocal directions total 10)
%e A220639 ..0..6..4....0..0..0....6..4..0....8..9..0....8..9..0....8..0..0....6..4..0
%e A220639 ..9..0..8....9..0..0....8..6..4....2..9..1....2..0..1....2..0..0....0..0..0
%e A220639 ..0..1..2....0..1..0....2..6..4....0..0..1....6..4..0....0..6..4....6..4..0
%p A220639 gf:= -(x^4-3*x^3+6*x^2+5*x-1)/((x-1)*(3*x^5-7*x^4+9*x^3+29*x^2+7*x-1)):
%p A220639 a:= n-> coeff(series(gf, x, n+1), x, n):
%p A220639 seq(a(n), n=0..30);  # _Alois P. Heinz_, Jun 03 2014
%t A220639 LinearRecurrence[{8,22,-20,-16,10,-3},{3,40,369,3755,37320,373177},30] (* _Harvey P. Dale_, Nov 17 2013 *)
%K A220639 nonn,easy
%O A220639 0,2
%A A220639 _R. H. Hardin_, Dec 17 2012