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A054420
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Number of connectable 3 X n binary matrices.
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2
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1, 3, 13, 87, 585, 3899, 25973, 173039, 1152849, 7680691, 51171485, 340922567, 2271346969, 15132518507, 100818201477, 671686589663, 4475014115745, 29814130048611, 198632300941357, 1323358787022391, 8816685256575721
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OFFSET
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1,2
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COMMENTS
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A connected (0,1) matrix is one where you can get from any black square, i.e., a1, to any other by chess king moves. A matrix is connectable if it is not connected, has rightmost column [1,0,1]' and becomes connected when any of [1,1,1]', [1,1,0]', [0,1,1]' or [0,1,0]' is appended.
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REFERENCES
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R. Levy and J. Shapiro, Uniqueness in paint-by-numbers puzzles, preprint, 2000.
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LINKS
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FORMULA
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a(n) = 7*a(n-1) - 3*a(n-2) + 5*a(n-3).
G.f.: x*(1+x)*(1-5*x)/(1 - 7*x + 3*x^2 - 5*x^3). - Colin Barker, Mar 08 2012
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MATHEMATICA
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CoefficientList[Series[(1+x)*(1-5*x)/(1-7*x+3*x^2-5*x^3), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 28 2012 *)
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PROG
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(Magma) I:=[1, 3, 13]; [n le 3 select I[n] else 7*Self(n-1)-3*Self(n-2)+5*Self(n-3): n in [1..25]]; // Vincenzo Librandi, Apr 28 2012
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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