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A225831
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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.
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5
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1, 72, 4224, 529920, 67133440, 8590786560, 1099516870656, 140737630961664, 18014399717441536, 2305843036057239552, 295147905471410601984, 37778931868592158801920, 4835703278531084466257920, 618970019643974367030804480, 79228162514282633467030142976, 10141204801826143708548100521984, 1298074214633711554847439528656896, 166153499473114560494025562738655232
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OFFSET
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0,2
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LINKS
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FORMULA
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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]
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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