A225833 Number of binary pattern classes in the (9,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, 272, 66048, 33632256, 17180262400, 8796137062400, 4503599962914816, 2305843036057239552, 1180591621026648948736, 604462909825456529211392, 309485009821644135887536128, 158456325028542467460946722816

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (544,-15872,-278528,8388608).

Formula

a(n) = 2^9*a(n-1) + 2^9*a(n-2) - (2^9)^2*a(n-3) - 2^(((9+1)/2)*n - 3)*(2^((9-1)/2)-1) with n>2, a(0)=1, a(1)=272, a(2)=66048.

a(n) = 2^(9n/2-1)*(2^(9n/2-1) + 2^(n/2-1) + 1) if n is even,

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

G.f.: (1-272*x-66048*x^2+2297856*x^3)/((1-32*x)*(1-512*x)*(1-512*x^2)). [Bruno Berselli, May 17 2013]

a(n) = 2^(5n-2)+2^(9n-2)+(34-(17-sqrt(2))*(1+(-1)^n))*sqrt(2)^(9n-5). [Bruno Berselli, May 17 2013]

Mathematica

LinearRecurrence[{544, -15872, -278528, 8388608}, {1, 272, 66048, 33632256}, 20] (* Bruno Berselli, May 17 2013 *)
CoefficientList[Series[(1 - 272 x - 66048 x^2 + 2297856 x^3) / ((1 - 32 x) (1 - 512 x) (1 - 512 x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Sep 04 2013 *)

Prog

(Magma) [2^(5*n-2)+2^(9*n-2)+(34-(17-Sqrt(2))*(1+(-1)^n))*Sqrt(2)^(9*n-5): n in [0..16]]; // Vincenzo Librandi, Sep 04 2013

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.

Sequence in context: A162009 A283789 A331655 * A198289 A198595 A028467

Adjacent sequences: A225830 A225831 A225832 * A225834 A225835 A225836

Keyword

nonn,easy

Author

Yosu Yurramendi, May 16 2013

Status

approved