A225833 - OEIS

%I #23 Sep 08 2022 08:46:05

%S 1,272,66048,33632256,17180262400,8796137062400,4503599962914816,

%T 2305843036057239552,1180591621026648948736,604462909825456529211392,

%U 309485009821644135887536128,158456325028542467460946722816

%N 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 Vincenzo Librandi, <a href="/A225833/b225833.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (544,-15872,-278528,8388608).

%F 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 a(n) = 2^(9n/2-1)*(2^(9n/2-1) + 2^(n/2-1) + 1) if n is even,

%F 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 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 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 LinearRecurrence[{544, -15872, -278528, 8388608}, {1, 272, 66048, 33632256}, 20] (* _Bruno Berselli_, May 17 2013 *)

%t 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 (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 A005418 is the number of binary pattern classes in the (1,n)-rectangular grid.

%Y A225826 to A225834  are the numbers of binary pattern classes in the (m,n)-rectangular grid, 1 < m < 11 .

%Y A225910 is the table of (m,n)-rectangular grids.

%K nonn,easy

%O 0,2

%A _Yosu Yurramendi_, May 16 2013