A137939 -id:A137939 - cp's OEIS frontend

A101363 In the interior of a regular 2n-gon with all diagonals drawn, the number of points where exactly three diagonals intersect.

0, 1, 8, 20, 60, 112, 208, 216, 480, 660, 864, 1196, 1568, 2250, 2464, 2992, 3924, 4332, 5160, 8148, 7040, 8096, 10560, 10600, 12064, 15552, 15288, 17052, 25320, 21080, 23360, 30360, 28288, 30940, 36288, 36852, 40128, 50076, 47120, 50840, 67620

Offset: 2

Graeme McRae, Dec 26 2004, revised Feb 23 2008, Feb 26 2008

nonn

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.
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.
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 points 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."

			a(6)=60 because inside a regular 12-gon there are 60 points (4 on each radius and 1 midway between radii) where exactly three diagonals intersect.

Seiichi Manyama, Table of n, a(n) for n = 2..10000 (terms 2..105 from Graeme McRae)
M. F. Hasler, Interactive illustration of A006561(n)
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG], 1995-2006, which has fewer typos than the SIAM version.
B. Poonen and M. Rubinstein, Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics, Vol. 11, pp. 135-156 (1998). [Copy on SIAM web site]
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, SIAM J. on Discrete Mathematics, Vol. 11, No. 1, 135-156 (1998). [Copy on B. Poonen's web site]
B. Poonen and M. Rubinstein, Mathematica programs for A006561 and related sequences
M. Rubinstein, Drawings for n=4,5,6,...
N. J. A. Sloane, Illustrations of a(8) and a(9)
N. J. A. Sloane, Summary table for vertices and regions in regular n-gon with all chords drawn, for n = 3..19. [V = total number of vertices (A007569), V_i (i>=2) = number of vertices where i lines cross (A292105, A292104, A101363); R = total number of cells or regions (A007678), R_i (i>=3) = number of regions with i edges (A331450, A062361, A067151).]
R. G. Wilson V, Illustration of a(10)
Index entry for Sequences formed by drawing all diagonals in regular polygon

Cf. A006561, A007678, A101364, A101365
A column of A292105.
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. A292104: number of 2-way intersections in the interior of a regular n-gon
Cf. A101364: number of 4-way intersections in the interior of a regular n-gon
Cf. A101365: number of 5-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.

A101364 In the interior of a regular n-gon with all diagonals drawn, the number of points where exactly four diagonals intersect.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 12, 0, 0, 0, 0, 0, 54, 0, 0, 0, 0, 0, 264, 0, 0, 0, 0, 0, 420, 0, 0, 0, 0, 0, 396, 0, 0, 0, 0, 0, 1134, 0, 0, 0, 0, 0, 1200, 0, 0, 0, 0, 0, 1296, 0, 0, 0, 0, 0, 3780, 0, 0, 0, 0, 0, 2310, 0, 0, 0, 0, 0, 2520, 0, 0, 0, 0, 0, 3276, 0, 0, 0, 0, 0, 3612, 0, 0, 0, 0, 0, 4050

Offset: 3

Graeme McRae, Dec 26 2004, revised Feb 23 2008

nonn

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)."

			a(18)=54 because inside a regular 18-gon there are 54 points where exactly four diagonals intersect.

Seiichi Manyama, Table of n, a(n) for n = 3..10000 (terms 3..210 from Graeme McRae)
Sequences formed by drawing all diagonals in regular polygon

A column of A292105.
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. A101365: number of 5-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.

A101365 In the interior of a regular n-gon with all diagonals drawn, the number of points where exactly five diagonals intersect.

Original entry on oeis.org

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

Offset: 3

Graeme McRae, Dec 26 2004, revised Feb 23 2008

nonn

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)."

			a(18)=54 because inside a regular 18-gon there are 54 points (3 on each radius) where exactly five diagonals intersect.

Seiichi Manyama, Table of n, a(n) for n = 3..10000 (terms 3..210 from Graeme McRae)
Sequences formed by drawing all diagonals in regular polygon

A column of A292105.
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.

A137938 Number of 4-way intersections in the interior of a regular 6n-gon.

Original entry on oeis.org

0, 12, 54, 264, 420, 396, 1134, 1200, 1296, 3780, 2310, 2520, 3276, 3612, 4050, 5088, 5712, 5724, 7182, 11400, 9072, 9372, 10626, 11088, 12600, 13260, 14094, 15960, 17052, 23220, 19530, 20928, 21384, 23052, 26250, 25704, 27972, 28956, 30186, 39600, 34440, 34524

Offset: 1

Graeme McRae, Feb 23 2008

nonn

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)."

			a(3)=54 because there are 54 points in the interior of an 18-gon at which exactly four diagonals intersect.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
B. Poonen and M. Rubinstein, The number of intersection points made by the diagonals of a regular polygon, arXiv:math/9508209 [math.MG]; some typos in the published version are corrected in the revisions from 2006.
Sequences formed by drawing all diagonals in regular polygon

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. A101365: number of 5-way intersections in the interior of a regular n-gon.
Cf. A137939: number of 5-way intersections in the interior of a regular 6n-gon

a(n) = A101364(6*n). - Seiichi Manyama, Jul 20 2024

More terms from Seiichi Manyama, Jul 20 2024

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A101363 In the interior of a regular 2n-gon with all diagonals drawn, the number of points where exactly three diagonals intersect.

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A101364 In the interior of a regular n-gon with all diagonals drawn, the number of points where exactly four diagonals intersect.

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A101365 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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A137938 Number of 4-way intersections in the interior of a regular 6n-gon.

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