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A132390 Number of binary pattern classes in the (2,n)-rectangular grid; two patterns are in same class if one of them can be obtained by reflection or rotation of the other one.

%I A132390 #41 Apr 14 2024 03:45:10
%S A132390 3,6,24,76,288,1072,4224,16576,66048,262912,1050624,4197376,16785408,
%T A132390 67121152,268468224,1073790976,4295098368,17180065792,68720001024,
%U A132390 274878693376,1099513724928,4398049656832,17592194433024,70368756760576
%N A132390 Number of binary pattern classes in the (2,n)-rectangular grid; two patterns are in same class if one of them can be obtained by reflection or rotation of the other one.
%C A132390 A005418 is the solution for the problem in the (1,n)-rectangular grid.
%C A132390 For n != 2, a(n) = 4^(n-1) + 2*A133572(n-1). - _Jon E. Schoenfield_, Aug 25 2009
%C A132390 A225826 is the same sequence, except a(2)=7. Here, 90-degree rotation is allowed, so a(2)=6. [_Yosu Yurramendi_, May 18 2013 - communicated by _Jon E. Schoenfield_]
%H A132390 Vincenzo Librandi, <a href="/A132390/b132390.txt">Table of n, a(n) for n = 1..1000</a>
%H A132390 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,-16).
%F A132390 For n != 2, a(n) = 4^(n-1) + 2^(n-2)*(3 + (n mod 2)). - _Jon E. Schoenfield_, Aug 25 2009
%F A132390 From _Colin Barker_, May 20 2013: (Start)
%F A132390 a(n) = 2^(-3+n)*(7 - (-1)^n + 2^(1+n)) for n > 2.
%F A132390 a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3), n >= 6.
%F A132390 G.f.: -x*(16*x^4 - 4*x^3 + 12*x^2 + 6*x - 3) / ((2*x-1)*(2*x+1)*(4*x-1)). (End)
%t A132390 CoefficientList[Series[-(16 x^4 - 4 x^3 + 12 x^2 + 6 x - 3) / ((2 x - 1) (2 x + 1) (4 x - 1)), {x, 0, 30}], x] (* _Vincenzo Librandi_, Sep 04 2013 *)
%t A132390 LinearRecurrence[{4,4,-16},{3,6,24,76,288},30] (* _Harvey P. Dale_, Sep 22 2016 *)
%o A132390 (Magma) I:=[3,6,24,76,288]; [n le 5 select I[n] else 4*Self(n-1)+4*Self(n-2)-16*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Sep 04 2013
%Y A132390 Cf. A005418, A034851.
%K A132390 nonn,easy
%O A132390 1,1
%A A132390 _Yosu Yurramendi_, Aug 26 2008
%E A132390 More terms from _Jon E. Schoenfield_, Aug 25 2009, corrected Aug 30 2009