A225827 Number of binary pattern classes in the (3,n)-rectangular grid: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

1, 6, 24, 168, 1120, 8640, 66816, 529920, 4212736, 33632256, 268713984, 2148630528, 17184194560, 137456517120, 1099579785216, 8796367749120, 70369826308096, 562954298720256, 4503616874348544, 36028866141093888, 288230651566489600, 2305844111946547200

Offset: 0

Author

Yosu Yurramendi, May 16 2013

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (12,-24,-96,256).

Crossrefs

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

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

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

Programs

Magma

I:=[1,6,24,168]; [n le 4 select I[n] else 12*Self(n-1)-24*Self(n-2)-96*Self(n-3)+256*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Sep 04 2013

Mathematica

LinearRecurrence[{12, -24, -96, 256}, {1, 6, 24, 168}, 20] (* Bruno Berselli, May 17 2013 *)
CoefficientList[Series[(1 - 6 x - 24 x^2 + 120 x^3) / ((1 - 4 x) (1 - 8 x) (1 - 8 x^2)), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 04 2013 *)

Formula

a(n) = 8*a(n-1) + 8*a(n-2) - 64*a(n-3) - 2^(2n-3) with n>2, with a(0)=1, a(1)=6, a(2)=24.

a(n) = 2^(3n/2-1)*(2^(3n/2-1) + 2^(n/2-1) + 1) if n is even,

a(n) = 2^((3*n-1)/2-1)*(2^((3*n-1)/2) + 2^((n-1)/2) + 3) if n is odd.

G.f.: (1-6*x-24*x^2+120*x^3)/((1-4*x)*(1-8*x)*(1-8*x^2)). [Bruno Berselli, May 17 2013]

Extensions

More terms from Vincenzo Librandi, Sep 04 2013