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A308305
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a(n) = s(n,n) + s(n,n-1) + s(n,n-2), where s(n,k) are the unsigned Stirling numbers of the first kind (see A132393).
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2
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1, 2, 6, 18, 46, 101, 197, 351, 583, 916, 1376, 1992, 2796, 3823, 5111, 6701, 8637, 10966, 13738, 17006, 20826, 25257, 30361, 36203, 42851, 50376, 58852, 68356, 78968, 90771, 103851, 118297, 134201, 151658, 170766, 191626, 214342, 239021, 265773, 294711
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OFFSET
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1,2
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COMMENTS
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Pairwise perpendicular bisectors divide the Euclidean plane into a maximum of a(n) regions. This maximum value a(n) occurs when no three points are collinear and no four points are concyclic in the plane, and with no perpendicular bisectors parallel or coinciding [Zaslavsky, Eq. (1.1)]. This count of regions in the plane is relevant for social science applications to voting preferences based on proximity to candidates on issues.
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REFERENCES
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T. Zaslavsky, Perpendicular dissections of space. Discrete Comput. Geom. 27 (2002), no. 3, 303-351.
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LINKS
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Alvaro Carbonero, Beth Anne Castellano, Gary Gordon, Charles Kulick, Karie Schmitz, and Brittany Shelton, Permutations of point sets in R_d, arXiv:2106.14140 [math.CO], 2021.
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FORMULA
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a(n) = s(n,n) + s(n,n-1) + s(n,n-2), where s(n,k) are the unsigned Stirling numbers of the first kind.
a(n) = (1/24)*(24 - 14*n + 21*n^2 - 10*n^3 + 3*n^4).
G.f.: x*(1 - 3*x + 6*x^2 - 2*x^3 + x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.
(End)
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MATHEMATICA
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Table[(1/24)(24 - 14 i + 21 i^2 - 10 i^3 + 3 i^4), {i, 40}]
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PROG
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(Magma) [(1/24)*(24 - 14*n + 21*n^2 - 10*n^3 + 3*n^4): n in [1..40]]; // Vincenzo Librandi, Jun 30 2019
(PARI) Vec(x*(1 - 3*x + 6*x^2 - 2*x^3 + x^4) / (1 - x)^5 + O(x^40)) \\ Colin Barker, Jun 30 2019
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CROSSREFS
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The unsigned Stirling numbers of the first kind s(n,k) are given in A132393.
The division of space formulation can be generalized to higher dimensions with use of A008275 by Good and Tideman's work.
The maximum number of regions generated by pairwise perpendicular bisectors on a sphere is given by A087645.
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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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