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%I #22 Sep 08 2022 08:46:05
%S 1,6,24,168,1120,8640,66816,529920,4212736,33632256,268713984,
%T 2148630528,17184194560,137456517120,1099579785216,8796367749120,
%U 70369826308096,562954298720256,4503616874348544,36028866141093888,288230651566489600,2305844111946547200
%N Number of binary pattern classes in the (3,n)-rectangular grid: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.
%H Vincenzo Librandi, <a href="/A225827/b225827.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (12,-24,-96,256).
%F a(n) = 8*a(n-1) + 8*a(n-2) - 64*a(n-3) - 2^(2n-3) with n>2, with a(0)=1, a(1)=6, a(2)=24.
%F a(n) = 2^(3n/2-1)*(2^(3n/2-1) + 2^(n/2-1) + 1) if n is even,
%F a(n) = 2^((3*n-1)/2-1)*(2^((3*n-1)/2) + 2^((n-1)/2) + 3) if n is odd.
%F G.f.: (1-6*x-24*x^2+120*x^3)/((1-4*x)*(1-8*x)*(1-8*x^2)). [_Bruno Berselli_, May 17 2013]
%t LinearRecurrence[{12, -24, -96, 256}, {1, 6, 24, 168}, 20] (* _Bruno Berselli_, May 17 2013 *)
%t CoefficientList[Series[(1 - 6 x - 24 x^2 + 120 x^3) / ((1 - 4 x) (1 - 8 x) (1 - 8 x^2)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 04 2013 *)
%o (Magma) I:=[1,6,24,168]; [n le 4 select I[n] else 12*Self(n-1)-24*Self(n-2)-96*Self(n-3)+256*Self(n-4): n in [1..30]]; // _Vincenzo Librandi_, Sep 04 2013
%Y A005418 is the number of binary pattern classes in the (1,n)-rectangular grid.
%Y A225826 to A225834 are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11.
%Y A225910 is the table of (m,n)-rectangular grids.
%K nonn,easy
%O 0,2
%A _Yosu Yurramendi_, May 16 2013
%E More terms from _Vincenzo Librandi_, Sep 04 2013
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