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A225828 Number of binary pattern classes in the (4,n)-rectangular grid: two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

%I A225828 #32 Sep 08 2022 08:46:05
%S A225828 1,10,76,1120,16576,263680,4197376,67133440,1073790976,17180262400,
%T A225828 274878693376,4398052802560,70368756760576,1125900007505920,
%U A225828 18014398710808576,288230377762324480,4611686021648613376,73786976320608010240,1180591620768950910976,18889465931890897715200
%N A225828 Number of binary pattern classes in the (4,n)-rectangular grid: two patterns are in the same class if one of them can be obtained by a reflection or 180-degree rotation of the other.
%H A225828 Vincenzo Librandi, <a href="/A225828/b225828.txt">Table of n, a(n) for n = 0..800</a>
%H A225828 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (16,16,-256).
%F A225828 a(n) = 16*a(n-1) + 16*a(n-2) - (16^2)*a(n-3) with n>2, a(0)=1, a(1)=10, a(2)=76.
%F A225828 a(n) = 2^(2n-3)*(2^(2n+1)-3*(-1)^n+9).
%F A225828 G.f.: (1-6*x-100*x^2)/((1-4*x)*(1+4*x)*(1-16*x)). [_Bruno Berselli_, May 16 2013]
%t A225828 Table[2^(2 n - 3) (2^(2 n + 1) - 3 (-1)^n + 9), {n, 0, 20}] (* _Bruno Berselli_, May 16 2013 *)
%t A225828 LinearRecurrence[{16, 16, -256}, {1, 10, 76}, 20] (* _Bruno Berselli_, May 17 2013 *)
%t A225828 CoefficientList[Series[(1 - 6 x - 100 x^2) / ((1 - 4 x) (1 + 4 x) (1 - 16 x)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Sep 04 2013 *)
%o A225828 (Magma) I:=[1, 10, 76]; [n le 3 select I[n] else 16*Self(n-1)+16*Self(n-2)-256*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Sep 04 2013
%Y A225828 A005418 is the number of binary pattern classes in the (1,n)-rectangular grid.
%Y A225828 A225826 to A225834 are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11.
%Y A225828 A225910 is the table of (m,n)-rectangular grids.
%K A225828 nonn,easy
%O A225828 0,2
%A A225828 _Yosu Yurramendi_, May 16 2013
%E A225828 More terms from _Vincenzo Librandi_, Sep 04 2013