A225828 Number of binary pattern classes in the (4,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.

1, 10, 76, 1120, 16576, 263680, 4197376, 67133440, 1073790976, 17180262400, 274878693376, 4398052802560, 70368756760576, 1125900007505920, 18014398710808576, 288230377762324480, 4611686021648613376, 73786976320608010240, 1180591620768950910976, 18889465931890897715200

Offset: 0

Author

Yosu Yurramendi, May 16 2013

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..800
• Index entries for linear recurrences with constant coefficients, signature (16,16,-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, 10, 76]; [n le 3 select I[n] else 16*Self(n-1)+16*Self(n-2)-256*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Sep 04 2013

Mathematica

Table[2^(2 n - 3) (2^(2 n + 1) - 3 (-1)^n + 9), {n, 0, 20}] (* Bruno Berselli, May 16 2013 *)
LinearRecurrence[{16, 16, -256}, {1, 10, 76}, 20] (* Bruno Berselli, May 17 2013 *)
CoefficientList[Series[(1 - 6 x - 100 x^2) / ((1 - 4 x) (1 + 4 x) (1 - 16 x)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 04 2013 *)

Formula

a(n) = 16*a(n-1) + 16*a(n-2) - (16^2)*a(n-3) with n>2, a(0)=1, a(1)=10, a(2)=76.

a(n) = 2^(2n-3)*(2^(2n+1)-3*(-1)^n+9).

G.f.: (1-6*x-100*x^2)/((1-4*x)*(1+4*x)*(1-16*x)). [Bruno Berselli, May 16 2013]

Extensions

More terms from Vincenzo Librandi, Sep 04 2013