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A241600
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Number of ways of arranging n lines in the (affine) plane.
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6
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OFFSET
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0,3
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COMMENTS
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This is in the affine plane, rather than the projective plane, so lines are either parallel or meet in one point.
Two arrangements are considered the same if one can be continuously changed to the other while keeping all lines straight, without changing the multiplicity of intersection points, and without a line passing through an intersection point. Turning over is also allowed.
a(n) might be called the size of the moduli space of n lines in the affine plane.
The subsequence giving the number of arrangements G_n of n lines in "general position" (with every two lines meeting in one point and every intersection point lying on exactly two lines) is given by A090338.
The moduli space of n points in the affine plane has been studied by several people (see for example Haiman and Miller, 2004; Martin, 2003). There is no direct connection with this problem, but these references are included for background information. - N. J. A. Sloane, Sep 13 2014
Lukas Finschi points out (email, Sep 19 2014) that a(n) = A063859(n)+1 for n <= 7 (but not for larger n). - N. J. A. Sloane, Sep 20 2014
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REFERENCES
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B. Grünbaum, Arrangements and Spreads. American Mathematical Society, Providence, RI, 1972, p. 4.
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LINKS
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FORMULA
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EXAMPLE
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Let P_n = n parallel lines, S_n = star of n lines through a point, G_n = n lines in general position, L = P_1 = S_1 = G_1 = a single line.
a(1) = 1: L.
a(2) = 2: P_2, S_2.
a(3) = 4: P_3, P_2 L, S_3, G_3.
See link for illustrations of first 5 terms.
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CROSSREFS
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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a(6) and a(7) from Lukas Finschi, Sep 19 2014
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STATUS
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approved
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