%I #20 Sep 08 2022 08:45:54
%S 8,64,400,2500,16100,103684,665252,4268356,27399292,175880644,
%T 1128941012,7246435876,46513697660,298563888100,1916431442740,
%U 12301251494596,78959676072668,506828955431044,3253250254953428,20882069005614436
%N Number of n X 3 binary matrices with no three 1's adjacent in a line diagonally or antidiagonally.
%C Column 3 of A181217.
%H R. H. Hardin, <a href="/A181214/b181214.txt">Table of n, a(n) for n = 1..300</a>
%H Robert Israel, <a href="/A181214/a181214.pdf">Maple-assisted proof of formula</a>
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (6,0,16, 21,-78,-32,0,-12,8).
%F Empirical: a(n) = 6*a(n-1) + 16*a(n-3) + 21*a(n-4) - 78*a(n-5) - 32*a(n-6) - 12*a(n-8) + 8*a(n-9).
%F Empirical g.f.: 4*x*(2 + 4*x + 4*x^2 - 7*x^3 - 23*x^4 - 9*x^5 - x^6 - 2*x^7 + 2*x^8) / ((1 - 6*x - 3*x^2 + 2*x^3)*(1 + 3*x^2 - 12*x^4 - 4*x^6)). - _Colin Barker_, Feb 22 2018
%F Empirical formula confirmed by _Robert Israel_, Apr 30 2018: see link.
%e Some avoided solutions for 4 X 3:
%e 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
%e 0 0 1 0 0 1 1 0 1 0 0 1 1 0 1 1 0 0 0 0 1
%e 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 0
%e 1 0 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 1 0 1
%p f:= gfun:-rectoproc({a(n) = 6*a(n-1) + 16*a(n-3) + 21*a(n-4) - 78*a(n-5) - 32*a(n-6) - 12*a(n-8) + 8*a(n-9),seq(a(i)=[8, 64, 400, 2500, 16100, 103684, 665252, 4268356, 27399292][i],i=1..9)},a(n),remember):
%p map(f, [$1..20]); # _Robert Israel_, Apr 30 2018
%t LinearRecurrence[{6, 0, 16, 21, -78, -32, 0, -12, 8}, {8, 64, 400, 2500, 16100, 103684, 665252, 4268356, 27399292}, 20] (* _Vincenzo Librandi_, May 01 2018 *)
%o (Magma) I:=[8,64,400,2500,16100,103684,665252,4268356,27399292]; [n le 9 select I[n] else 6*Self(n-1)+16*Self(n-3)+21*Self(n-4)-78*Self(n-5) -32*Self(n-6)-12*Self(n-8)+8*Self(n-9): n in [1..25]]; // _Vincenzo Librandi_, May 01 2018
%Y Cf. A181217.
%K nonn
%O 1,1
%A _R. H. Hardin_, Oct 10 2010