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Search: id:A101365
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| A101365 |
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In the interior of a regular n-gon with all diagonals drawn, the number of points where exactly five diagonals intersect. |
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5
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| 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 180, 0, 0, 0, 0, 0, 216, 0, 0, 0, 0, 0, 546, 0, 0, 0, 0, 0, 336, 0, 0, 0, 0, 0, 648, 0, 0, 0, 0, 0, 720, 0, 0, 0, 0, 0, 990, 0, 0, 0, 0, 0, 936, 0, 0, 0, 0, 0, 1404, 0, 0, 0, 0, 0, 2352, 0, 0, 0, 0, 0, 1890, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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3,16
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COMMENT
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When n is odd, there are no intersections in the interior of an n-gon where more than 2 diagonals meet.
When n is not a multiple of 6, there are no intersections in the interior of an n-gon where more than 3 diagonals meet except the center.
When n is not a multiple of 30, there are no intersections in the interior of an n-gon where more than 5 diagonals meet except the center.
I checked the following conjecture up to n=210: "An n-gon with n=30k has 5n points where 6 or 7 diagonals meet and no interior point other than the center where more than 7 diagonals meet; If k is odd, then 6 diagonals meet in each of 4n points and 7 diagonals meet in each of n points; If k is even, then no groups of exactly 6 diagonals meet in a point, while exactly 7 diagonals meet in each of 5n points (all points interior excluding the center)."
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LINKS
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Graeme McRae (g_m(AT)mcraefamily.com), Feb 23 2008, Table of n, a(n) for n = 3..210
Sequences formed by drawing all diagonals in regular polygon
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EXAMPLE
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a(18)=54 because inside a regular 18-gon there are 54 points (3 on each radius) where exactly five diagonals intersect.
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CROSSREFS
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Cf. A006561, A007678.
Cf. A000332: C(n, 4) = number of intersection points of diagonals of convex n-gon.
Cf. A006561: number of intersections of diagonals in the interior of regular n-gon
Cf. A101363: number of 3-way intersections in the interior of a regular 2n-gon
Cf. A101364: number of 4-way intersections in the interior of a regular n-gon
Cf. A137938: number of 4-way intersections in the interior of a regular 6n-gon
Cf. A137939: number of 5-way intersections in the interior of a regular 6n-gon
Sequence in context: A145332 A087530 A125037 this_sequence A022080 A033688 A058283
Adjacent sequences: A101362 A101363 A101364 this_sequence A101366 A101367 A101368
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KEYWORD
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nonn
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AUTHOR
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Graeme McRae (g_m(AT)mcraefamily.com), Dec 26 2004, revised Feb 23 2008
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