A146213 -id:A146213 - cp's OEIS frontend

A146212 Number of intersection points of all lines through pairs of vertices of a regular n-gon.

3, 5, 15, 37, 91, 145, 333, 471, 891, 901, 1963, 2185, 3795, 3969, 6681, 5563, 10963, 11141, 17031, 17293, 25323, 21913, 36325, 36479, 50571, 50485, 68643, 51661, 91171, 90753, 118833, 118355, 152355, 139861, 192511, 191445, 240123, 238481

Offset: 3

T. D. Noe, Oct 28 2008

This includes intersection points outside of the n-gon. Note that for odd n, n divides a(n); for even n, n divides a(n)-1. For odd n, it appears that a(n)=n*(n^3-7*n^2+15*n-1)/8.

That formula for odd n is correct: see the Sidorenko link. - N. J. A. Sloane, Sep 12 2021

			a(5)=15 because there are 5 points inside the pentagon, 5 points on the pentagon and five points outside of the pentagon.

Jon E. Schoenfield, Table of n, a(n) for n = 3..100
T. D. Noe, Pentagon Illustrated
J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.
Scott R. Shannon, Image for n = 3. In this and other images the dots showing the regular n-gon's vertices are slightly larger and circled with white for clarity. The dot color key is at the top-left of the image.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5.
Scott R. Shannon, Image for n = 6.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.
Scott R. Shannon, Image for n = 9.
Scott R. Shannon, Image for n = 10.
Scott R. Shannon, Image for n = 11.
Scott R. Shannon, Image for n = 12.
Alexander Sidorenko, Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857.

Cf. A006561, A007569, A146213.

Bisection: A347319, A347321.

There is a formula for odd n: see Comment section and the Sidorenko link. - N. J. A. Sloane, Sep 12 2021

More terms from Jon E. Schoenfield, Nov 10 2008

Definition clarified by N. J. A. Sloane, Jun 06 2025

A344311 Number of polygons formed outside a regular n-gon when every pair of vertices of the n-gon are joined by an infinite line.

Original entry on oeis.org

0, 0, 0, 0, 5, 18, 49, 96, 198, 320, 550, 708, 1235, 1582, 2415, 2912, 4284, 4518, 7068, 8080, 11025, 12430, 16445, 16992, 23650, 26000, 32994, 35896, 44863, 42090, 59675, 64064, 77880, 83232, 99960, 102132, 126429, 133950, 157833, 166560, 194750, 191310, 237790, 249480, 287595, 301070

Offset: 1

Scott R. Shannon, Jun 01 2021

nonn

See A344857 for other examples and images of the polygons.

Only finite polygons are counted. - N. J. A. Sloane, Jun 11 2021

			a(5) = 5 as the five connected vertices of a regular pentagon form five triangles outside the pentagon, along with sixteen polygons on the inside.

Cf. A344857 (total number of polygons), A007678 (number of polygons inside the n-gon), A345025 (total number of polygons and open regions), A344938, A344857, A344866, A146213 (number of vertices).

For odd n, a(n) = (2*n^4 - 15*n^3 + 34*n^2 - 21*n)/24 (see A344857).

a(n) = A344857(n) - A007678(n).

Edited by N. J. A. Sloane, Sep 12 2021

A211382 Number of intersections of diagonals in the exterior of a regular n-gon.

Original entry on oeis.org

0, 0, 0, 0, 7, 24, 63, 110, 242, 360, 650, 812, 1425, 1680, 2737, 2718, 4788, 5200, 7812, 8272, 12075, 11328, 17875, 18486, 25542, 26264, 35438, 29070, 47957, 48800, 63525, 64362, 82600, 77940, 105672, 106552, 133263, 134200, 165927, 149478, 204250, 205128, 248850, 249596, 300377

Offset: 3

Martin Renner, Feb 07 2013

nonn

Robin Visser, Table of n, a(n) for n = 3..95
Bjorn Poonen and Michael Rubinstein, The Number of Intersection Points Made by the Diagonals of a Regular Polygon, SIAM J. Discrete Mathematics 11 (1998), nr. 1, pp. 135-156; doi: 10.1137/S0895480195281246; arXiv: math.MG/9508209.
Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals.

Cf. A006561, A146213, A211383.

def a(n):
    K = CyclotomicField(n); z = K.gen(); S = set()
    for i in range(n):
        for j in range(i+2, n):
            for k in range(j+1, n):
                for l in range(k+2, n+i):
                    x = (z^(i-j)-z^(j-i))*(z^l-z^k)-(z^(k-l)-z^(l-k))*(z^j-z^i)
                    y = (z^-j-z^-i)*(z^l-z^k)-(z^-l-z^-k)*(z^j-z^i)
                    if (not y.is_zero()): S.add(x/y)
    return len(S)  # Robin Visser, Jul 29 2024

a(n) = 1/24*n*(n-3)*(n-5)*(2*n-11) for n odd

a(17)-a(29) from Martin Renner, Feb 24 2013

More terms from Robin Visser, Jul 29 2024

A383462 Triangle read by rows: T(n,k) (n >= 3, 2 <= k <= n-1) = number of vertices where k lines cross in the planar graph formed when every pair of vertices of a regular n-gon are joined by an infinite line.

Original entry on oeis.org

3, 1, 4, 10, 0, 5, 30, 1, 0, 6, 84, 0, 0, 0, 7, 120, 16, 1, 0, 0, 8, 324, 0, 0, 0, 0, 0, 9, 420, 40, 0, 1, 0, 0, 0, 10, 880, 0, 0, 0, 0, 0, 0, 0, 11, 708, 156, 24, 0, 1, 0, 0, 0, 0, 12, 1950, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 1890, 280, 0, 0, 0, 1, 0, 0, 0, 0, 0, 14

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Jun 07 2025

For other illustrations see A146212, A344857, A292105.

			Triangle begins:
   3;
   1, 4;
   10, 0, 5;
   30, 1, 0, 6;
   84, 0, 0, 0, 7;
   120, 16, 1, 0, 0, 8;
   324, 0, 0, 0, 0, 0, 9;
   420, 40, 0, 1, 0, 0, 0, 10;
   880, 0, 0, 0, 0, 0, 0, 0, 11;
   708, 156, 24, 0, 1, 0, 0, 0, 0, 12;
   1950, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13;
   1890, 280, 0, 0, 0, 1, 0, 0, 0, 0, 0, 14;
   3780, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 15;
   3408, 544, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 16;
   6664, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 17;
   4572, 756, 108, 108, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 18;
   10944, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 19;
   9840, 1280, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 20;
   .
   .
See the attached table for rows 3 to 100.
For n = 8, we may classify the vertices by degree and according to whether they are outside, on, or inside the octagon:
                V2      V3      V4      V5      V6      V7
----------------------------------------------------------
   outside      80      8
   on           0       0       0       0       0       8
   inside       40      8       1       0       0       0
----------------------------------------------------------
   totals       120     16      1       0       0       8
----------------------------------------------------------
   Grand total: 145 = A146212(8)
In general, for n >= 3, the counts for inside the defining polygon are given by row n of A292105, the total number on or inside the polygon by A007569, and the number outside by A146213.

Scott R. Shannon, Table of n, a(n) for n = 3..4853
Scott R. Shannon, Formatted table for rows 3 to 100.
Scott R. Shannon, Image of the 5-gon.
Scott R. Shannon, Image of the 6-gon.
Scott R. Shannon, Image of the 7-gon.
Scott R. Shannon, Image of the 8-gon.

Row sums are A146212.

Cf. A007569, A146213, A292105, A344857.

cp's OEIS Frontend

A146212 Number of intersection points of all lines through pairs of vertices of a regular n-gon.

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A344311 Number of polygons formed outside a regular n-gon when every pair of vertices of the n-gon are joined by an infinite line.

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A211382 Number of intersections of diagonals in the exterior of a regular n-gon.

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A383462 Triangle read by rows: T(n,k) (n >= 3, 2 <= k <= n-1) = number of vertices where k lines cross in the planar graph formed when every pair of vertices of a regular n-gon are joined by an infinite line.

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