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A225829 Number of binary pattern classes in the (5,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 A225829 #24 Sep 08 2022 08:46:05
%S A225829 1,20,288,8640,263680,8407040,268517376,8590786560,274882625536,
%T A225829 8796137062400,281475261923328,9007201737768960,288230393868451840,
%U A225829 9223372185031147520,295147906296044322816,9444732974878980833280,302231454974575793668096,9671406557490978467348480
%N A225829 Number of binary pattern classes in the (5,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 A225829 Vincenzo Librandi, <a href="/A225829/b225829.txt">Table of n, a(n) for n = 0..600</a>
%H A225829 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (40,-224,-1280,8192).
%F A225829 a(n) = 32*a(n-1) + 32*a(n-2) - 1024*a(n-3)- 2^(3n - 3)*3 with n>2, a(0)=1, a(1)=20, a(2)=288.
%F A225829 a(n) = 2^(5n/2-1)*(2^(5n/2-1) + 2^(n/2-1) + 1) if n is even,
%F A225829 a(n) = 2^((5n-1)/2-1)*(2^((5n-1)/2) + 2^((n-1)/2) + 5) if n is odd.
%F A225829 G.f.: (1-20*x-288*x^2+2880*x^3)/((1-8*x)*(1-32*x)*(1-32*x^2)). [_Bruno Berselli_, May 17 2013]
%t A225829 LinearRecurrence[{40,-224,-1280,8192}, {1, 20, 288, 8640}, 20] (* _Bruno Berselli_, May 17 2013 *)
%t A225829 CoefficientList[Series[(1 - 20 x - 288 x^2 + 2880 x^3) / ((1 - 8 x) (1 - 32 x) (1 - 32 x^2)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Sep 04 2013 *)
%o A225829 (Magma) I:=[1, 20, 288, 8640]; [n le 4 select I[n] else 40*Self(n-1)-224*Self(n-2)-1280*Self(n-3)+8192*Self(n-4): n in [1..25]]; // _Vincenzo Librandi_, Sep 04 2013
%Y A225829 A005418 is the number of binary pattern classes in the (1,n)-rectangular grid.
%Y A225829 A225826 to A225834  are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11 .
%Y A225829 A225910 is the table of (m,n)-rectangular grids.
%K A225829 nonn,easy
%O A225829 0,2
%A A225829 _Yosu Yurramendi_, May 16 2013