A225831 Number of binary pattern classes in the (7,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, 72, 4224, 529920, 67133440, 8590786560, 1099516870656, 140737630961664, 18014399717441536, 2305843036057239552, 295147905471410601984, 37778931868592158801920, 4835703278531084466257920, 618970019643974367030804480, 79228162514282633467030142976, 10141204801826143708548100521984, 1298074214633711554847439528656896, 166153499473114560494025562738655232

Offset: 0

Author

Yosu Yurramendi, May 16 2013

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (144,-1920,-18432,262144).

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:=[1,72,4224,529920]; [n le 4 select I[n] else 144*Self(n-1)-1920*Self(n-2)-18432*Self(n-3)+262144*Self(n-4): n in [1..20]]; // Vincenzo Librandi, Sep 04 2013

Mathematica

LinearRecurrence[{144, -1920, -18432, 262144}, {1, 72, 4224, 529920}, 20] (* Bruno Berselli, May 17 2013 *)
CoefficientList[Series[(1 - 72 x - 4224 x^2 + 78336 x^3) / ((1 - 16 x) (1 - 128 x) (1 - 128 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 04 2013 *)

Formula

a(n) = (2^7)*a(n-1) + (2^7)*a(n-2) - ((2^7)^2)*a(n-3) - 2^(4n-3)*7 with n>2, a(0)=1, a(1)=72, a(2)=4224.

a(n) = 2^(7n/2-1)*(2^(7n/2-1) + 2^(n/2-1) + 1) if n is even,

a(n) = 2^((7n-1)/2-1)*(2^((7n-1)/2) + 2^((n-1)/2) + 9) if n is odd.

G.f.: (1-72*x-4224*x^2+78336*x^3)/((1-16*x)*(1-128*x)*(1-128*x^2)). [Bruno Berselli, May 17 2013]