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
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).
Crossrefs
Programs
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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
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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 *)
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]