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A225830 Number of binary pattern classes in the (6,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 A225830 #27 Sep 08 2022 08:46:05
%S A225830 1,36,1072,66816,4197376,268517376,17180065792,1099516870656,
%T A225830 70368756760576,4503599962914816,288230376957018112,
%U A225830 18446744095184388096,1180591620768950910976,75557863727288712953856,4835703278461815233708032,309485009821433029655003136,19807040628566295504618520576,1267650600228235030996237418496
%N A225830 Number of binary pattern classes in the (6,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 A225830 Vincenzo Librandi, <a href="/A225830/b225830.txt">Table of n, a(n) for n = 0..500</a>
%H A225830 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (64,64,-4096).
%F A225830 a(n) = 64*a(n-1) + 64*a(n-2) - (64^2)*a(n-3) with n>2, a(0)=1, a(1)=36, a(2)=1072.
%F A225830 a(n) = 2^(3n-3)*(2^(3n+1)-(2^3-1)*(-1)^n+2^3+5) = 8^(n-1)*(2^(3n+1)-7*(-1)^n+13).
%F A225830 G.f.: (1-28*x-1296*x^2)/((1-8*x)*(1+8*x)*(1-64*x)).
%t A225830 LinearRecurrence[{64, 64, -4096}, {1, 36, 1072}, 20] (* _Bruno Berselli_, May 17 2013 *)
%t A225830 CoefficientList[Series[(1 - 28 x - 1296 x^2) / ((1 - 8 x) (1 + 8 x) (1 - 64 x)), {x, 0, 20}], x] (* _Vincenzo Librandi_, Sep 04 2013 *)
%o A225830 (Magma) I:=[1,36,1072]; [n le 3 select I[n] else 64*Self(n-1)+64*Self(n-2)-4096*Self(n-3): n in [1..25]]; // _Vincenzo Librandi_, Sep 04 2013
%Y A225830 A005418 is the number of binary pattern classes in the (1,n)-rectangular grid.
%Y A225830 A225826 to A225834  are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11 .
%Y A225830 A225910 is the table of (m,n)-rectangular grids.
%K A225830 nonn,easy
%O A225830 0,2
%A A225830 _Yosu Yurramendi_, May 16 2013