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.

3, 6, 24, 76, 288, 1072, 4224, 16576, 66048, 262912, 1050624, 4197376, 16785408, 67121152, 268468224, 1073790976, 4295098368, 17180065792, 68720001024, 274878693376, 1099513724928, 4398049656832, 17592194433024, 70368756760576

Offset: 1

Author

Yosu Yurramendi, Aug 26 2008

Keywords

• nonn
• easy

Comments

A005418 is the solution for the problem in the (1,n)-rectangular grid.

For n != 2, a(n) = 4^(n-1) + 2*A133572(n-1). - Jon E. Schoenfield, Aug 25 2009

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]

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,4,-16).

Crossrefs

Cf. A005418, A034851.

Programs

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

Mathematica

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 *)
LinearRecurrence[{4,4,-16},{3,6,24,76,288},30] (* Harvey P. Dale, Sep 22 2016 *)

Formula

For n != 2, a(n) = 4^(n-1) + 2^(n-2)*(3 + (n mod 2)). - Jon E. Schoenfield, Aug 25 2009

From Colin Barker, May 20 2013: (Start)

a(n) = 2^(-3+n)*(7 - (-1)^n + 2^(1+n)) for n > 2.

a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3), n >= 6.

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)

Extensions

More terms from Jon E. Schoenfield, Aug 25 2009, corrected Aug 30 2009