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A225826
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Number of binary pattern classes in the (2,n)-rectangular grid: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.
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23
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1, 3, 7, 24, 76, 288, 1072, 4224, 16576, 66048, 262912, 1050624, 4197376, 16785408, 67121152, 268468224, 1073790976, 4295098368, 17180065792, 68720001024, 274878693376, 1099513724928, 4398049656832, 17592194433024, 70368756760576
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OFFSET
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0,2
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LINKS
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Vincent Pilaud and V. Pons, Permutrees, arXiv preprint arXiv:1606.09643 [math.CO], 2016.
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FORMULA
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a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3) with n>2, a(0)=1, a(1)=3, a(2)=7 (communicated by Jon E. Schoenfield).
a(n) = 2^(n-3)*(2^(n+1) - (-1)^n + 7).
G.f.: (1-x-9*x^2)/((1-2*x)*(1+2*x)*(1-4*x)).
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MATHEMATICA
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LinearRecurrence[{4, 4, -16}, {1, 3, 7}, 30] (* Bruno Berselli, May 17 2013 *)
CoefficientList[Series[(1 - x - 9 x^2) / ((1 - 2 x) (1 + 2 x) (1 - 4 x)), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 03 2013 *)
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PROG
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(Magma) [2^(n-3)*(2^(n+1)-(-1)^n+7): n in [0..25]]; // Vincenzo Librandi, Sep 03 2013
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CROSSREFS
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Cf. A005418 = 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, A132390 is the sequence when the 90-degree rotation for pattern equivalence is allowed. So, only a(2) is different (communicated by Jon E. Schoenfield). See A054247 for (n,n)-grids.
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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