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A225831 Number of binary pattern classes in the (7,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 A225831 #23 Sep 08 2022 08:46:05
%S A225831 1,72,4224,529920,67133440,8590786560,1099516870656,140737630961664,
%T A225831 18014399717441536,2305843036057239552,295147905471410601984,
%U A225831 37778931868592158801920,4835703278531084466257920,618970019643974367030804480,79228162514282633467030142976,10141204801826143708548100521984,1298074214633711554847439528656896,166153499473114560494025562738655232
%N A225831 Number of binary pattern classes in the (7,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 A225831 Vincenzo Librandi, <a href="/A225831/b225831.txt">Table of n, a(n) for n = 0..400</a>
%H A225831 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (144,-1920,-18432,262144).
%F A225831 a(n) = (2^7)*a(n-1) + (2^7)*a(n-2) - ((2^7)^2)*a(n-3) - 2^(4n-3)*7 with n>2, a(0)=1, a(1)=72, a(2)=4224.
%F A225831 a(n) = 2^(7n/2-1)*(2^(7n/2-1) + 2^(n/2-1) + 1) if n is even,
%F A225831 a(n) = 2^((7n-1)/2-1)*(2^((7n-1)/2) + 2^((n-1)/2) + 9) if n is odd.
%F A225831 G.f.: (1-72*x-4224*x^2+78336*x^3)/((1-16*x)*(1-128*x)*(1-128*x^2)). [_Bruno Berselli_, May 17 2013]
%t A225831 LinearRecurrence[{144, -1920, -18432, 262144}, {1, 72, 4224, 529920}, 20] (* _Bruno Berselli_, May 17 2013 *)
%t A225831 CoefficientList[Series[(1 - 72 x - 4224 x^2 + 78336 x^3) / ((1 - 16 x) (1 - 128 x) (1 - 128 x^2)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Sep 04 2013 *)
%o A225831 (Magma) I:=[1,72,4224,529920]; [n le 4 select I[n] else 144*Self(n-1)-1920*Self(n-2)-18432*Self(n-3)+262144*Self(n-4): n in [1..20]]; // _Vincenzo Librandi_, Sep 04 2013
%Y A225831 A005418 is the number of binary pattern classes in the (1,n)-rectangular grid.
%Y A225831 A225826 to A225834  are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11 .
%Y A225831 A225910 is the table of (m,n)-rectangular grids.
%K A225831 nonn,easy
%O A225831 0,2
%A A225831 _Yosu Yurramendi_, May 16 2013