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A345459
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Number of polygons formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.
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8
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0, 4, 80, 568, 2024, 6052, 12144, 26976, 45024, 76724, 116840, 191128, 245976, 388452, 501888, 661476, 870168, 1199724, 1402096, 1911384, 2188320, 2739280, 3371264, 4224288, 4617224, 5801372, 6780568
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OFFSET
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0,2
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COMMENTS
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The width/height of the entire figure grows as ~ 2*n^3 for large n. See the Formula section below.
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LINKS
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Table of n, a(n) for n=0..26.
Scott R. Shannon, Image for n = 2. In this and other images the square's points are highlighted as white dots while the outer open regions, which are not counted, are darkened. The key for the edge-number coloring is shown at the top-left of the image.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
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FORMULA
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a(n) = A345650(n) - A345649(n) + 1.
Assuming the square is of size n x n centered on the origin the x (or y) offset for the eight outermost vertices is n^3 - 2*n^2 + 3*n/2, which have a corresponding y (or x) offset of n^2 - 3*n/2 + 1. The total distance from the origin of these vertices is sqrt(n^6 - 4*n^5 + 8*n^4 - 9*n^3 + 13*n^2/2 - 3*n + 1).
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EXAMPLE
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a(2) = 80. Connecting the 8 perimeter points results in the creation of forty-eight 3-gons and eight 4-gons inside the square while creating twenty-four 3-gons outside the square, giving eighty polygons in total. See the linked images.
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CROSSREFS
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Cf. A255011 (number inside the square), A345648 (number outside the square), A345649 (number of vertices), A345650 (number of edges), A344993, A344857, A092098, A007678.
Sequence in context: A192790 A211152 A093854 * A269146 A192834 A054322
Adjacent sequences: A345456 A345457 A345458 * A345460 A345461 A345462
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KEYWORD
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nonn,more
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AUTHOR
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Scott R. Shannon and N. J. A. Sloane, Jun 20 2021
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STATUS
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approved
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