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A225833 Number of binary pattern classes in the (9,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 A225833 #23 Sep 08 2022 08:46:05
%S A225833 1,272,66048,33632256,17180262400,8796137062400,4503599962914816,
%T A225833 2305843036057239552,1180591621026648948736,604462909825456529211392,
%U A225833 309485009821644135887536128,158456325028542467460946722816
%N A225833 Number of binary pattern classes in the (9,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 A225833 Vincenzo Librandi, <a href="/A225833/b225833.txt">Table of n, a(n) for n = 0..200</a>
%H A225833 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (544,-15872,-278528,8388608).
%F A225833 a(n) = 2^9*a(n-1) + 2^9*a(n-2) - (2^9)^2*a(n-3) - 2^(((9+1)/2)*n - 3)*(2^((9-1)/2)-1) with n>2, a(0)=1, a(1)=272, a(2)=66048.
%F A225833 a(n) = 2^(9n/2-1)*(2^(9n/2-1) + 2^(n/2-1) + 1) if n is even,
%F A225833 a(n) = 2^((9n-1)/2-1)*(2^((9n-1)/2) + 2^((n-1)/2) + 2^((9-1)/2) + 1) if n is odd.
%F A225833 G.f.: (1-272*x-66048*x^2+2297856*x^3)/((1-32*x)*(1-512*x)*(1-512*x^2)). [_Bruno Berselli_, May 17 2013]
%F A225833 a(n) = 2^(5n-2)+2^(9n-2)+(34-(17-sqrt(2))*(1+(-1)^n))*sqrt(2)^(9n-5). [_Bruno Berselli_, May 17 2013]
%t A225833 LinearRecurrence[{544, -15872, -278528, 8388608}, {1, 272, 66048, 33632256}, 20] (* _Bruno Berselli_, May 17 2013 *)
%t A225833 CoefficientList[Series[(1 - 272 x - 66048 x^2 + 2297856 x^3) / ((1 - 32 x) (1 - 512 x) (1 - 512 x^2)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Sep 04 2013 *)
%o A225833 (Magma) [2^(5*n-2)+2^(9*n-2)+(34-(17-Sqrt(2))*(1+(-1)^n))*Sqrt(2)^(9*n-5): n in [0..16]]; // _Vincenzo Librandi_, Sep 04 2013
%Y A225833 A005418 is the number of binary pattern classes in the (1,n)-rectangular grid.
%Y A225833 A225826 to A225834  are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11 .
%Y A225833 A225910 is the table of (m,n)-rectangular grids.
%K A225833 nonn,easy
%O A225833 0,2
%A A225833 _Yosu Yurramendi_, May 16 2013