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A105638
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Maximum number of intersections in self-intersecting n-gon.
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4
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0, 1, 5, 7, 14, 17, 27, 31, 44, 49, 65, 71, 90, 97, 119, 127, 152, 161, 189, 199, 230, 241, 275, 287, 324, 337, 377, 391, 434, 449, 495, 511, 560, 577, 629, 647, 702, 721, 779, 799, 860, 881, 945, 967, 1034, 1057, 1127, 1151, 1224, 1249, 1325, 1351, 1430, 1457
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OFFSET
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3,3
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COMMENTS
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REFERENCES
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B. Grünbaum, Selfintersections of Polygons, Geombinatorics, Volume VIII 4 (1998), pp. 37-45.
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LINKS
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FORMULA
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a(n) = n(n-3)/2 if n odd, n(n-4)/2+1 if n even.
a(n) = a(n-1) + 2a(n-2) - 2a(n-3) - a(n-4) + a(n-5).
G.f.: x^4*(1+4*x-x^3)/((1+x)^2*(1-x)^3). [Colin Barker, Jan 31 2012]
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EXAMPLE
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The self-intersecting pentagon with the largest number of intersections is the star polygon {5/2} (pentacle}, with 5 intersections, hence a(5) = 5.
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MATHEMATICA
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LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 5, 7, 14}, 54] (* or *)
DeleteCases[CoefficientList[Series[x^4*(1 + 4 x - x^3)/((1 + x)^2*(1 - x)^3), {x, 0, 56}], x], 0] (* Michael De Vlieger, Jul 10 2020 *)
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PROG
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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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STATUS
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approved
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