A146212 -id:A146212 - cp's OEIS frontend

A344907 Number of polygon edges formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

0, 3, 30, 189, 684, 1815, 3978, 7665, 13464, 22059, 34230, 50853, 72900, 101439, 137634, 182745, 238128, 305235, 385614, 480909, 592860, 723303, 874170, 1047489, 1245384, 1470075, 1723878, 2009205, 2328564, 2684559, 3079890, 3517353, 3999840, 4530339, 5111934, 5747805, 6441228, 7195575

Offset: 1

Scott R. Shannon, Jun 02 2021

This sequences gives the number of polygon edges formed when connecting every pair of vertices of a regular polygon, with an odd number of vertices, by an infinite line.
A bisection of A344899. - N. J. A. Sloane, Sep 12 2021
See A344857 for other examples and images of the polygons.

			a(3) = 30 as the five connected vertices form a pentagon with fives lines along the pentagon's edges, fifteen lines inside forming eleven polygons, and ten lines outside forming another five triangles. In all these sixteen polygons form thirty edges. Twenty infinite edges between the outer unbounded regions are also formed.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A344899 (number of edges for all n-gons), A344866 (number of polygon), A146212, A344857, A344311, A007678, A331450, A344938.

A347319 a(n) = (2n+1)(n^3-2*n^2+n+1).

Original entry on oeis.org

1, 3, 15, 91, 333, 891, 1963, 3795, 6681, 10963, 17031, 25323, 36325, 50571, 68643, 91171, 118833, 152355, 192511, 240123, 296061, 361243, 436635, 523251, 622153, 734451, 861303, 1003915, 1163541, 1341483, 1539091, 1757763, 1998945, 2264131, 2554863, 2872731, 3219373, 3596475, 4005771

Offset: 0

N. J. A. Sloane, Sep 12 2021

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

A bisection of A146212, analogous to A344866 and A344907.

a(n)=(2*n+1)*(n^3-2*n^2+n+1) \\ Charles R Greathouse IV, Oct 21 2022

def A347319(n): return n*(n**2*(2*n - 3) + 3) + 1 # Chai Wah Wu, Sep 12 2021

From Chai Wah Wu, Sep 12 2021: (Start)
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 4.
G.f.: (-3*x^4 - 36*x^3 - 10*x^2 + 2*x - 1)/(x - 1)^5. (End)

A357235 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of vertices in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts.

Original entry on oeis.org

3, 6, 4, 15, 8, 5, 30, 20, 10, 6, 51, 40, 25, 12, 7, 66, 68, 50, 30, 14, 8, 111, 88, 85, 60, 35, 16, 9, 150, 148, 130, 102, 70, 40, 18, 10, 171, 168, 185, 156, 119, 80, 45, 20, 11, 246, 260, 250, 222, 182, 136, 90, 50, 22, 12, 303, 296, 325, 300, 259, 208, 153, 100, 55, 24, 13

Offset: 3

Scott R. Shannon, Sep 19 2022

Conjecture: the only n-gons that contain non-simple intersections are the 3-gon (triangle), 4-gon (square), and 6-gon (hexagon).

			The table begins:
   3,  6, 15,  30,  51,  66, 111, 150, 171,  246,  303,  312,  435,  510,  543, ...
   4,  8, 20,  40,  68,  88, 148, 168, 260,  296,  404,  436,  580,  632,  788, ...
   5, 10, 25,  50,  85, 130, 185, 250, 325,  410,  505,  610,  725,  850,  985, ...
   6, 12, 30,  60, 102, 156, 222, 300, 390,  468,  606,  708,  870, 1020, 1152, ...
   7, 14, 35,  70, 119, 182, 259, 350, 455,  574,  707,  854, 1015, 1190, 1379, ...
   8, 16, 40,  80, 136, 208, 296, 400, 520,  656,  808,  976, 1160, 1360, 1576, ...
   9, 18, 45,  90, 153, 234, 333, 450, 585,  738,  909, 1098, 1305, 1530, 1773, ...
  10, 20, 50, 100, 170, 260, 370, 500, 650,  820, 1010, 1220, 1450, 1700, 1970, ...
  11, 22, 55, 110, 187, 286, 407, 550, 715,  902, 1111, 1342, 1595, 1870, 2167, ...
  12, 24, 60, 120, 204, 312, 444, 600, 780,  984, 1212, 1464, 1740, 2040, 2364, ...
  13, 26, 65, 130, 221, 338, 481, 650, 845, 1066, 1313, 1586, 1885, 2210, 2561, ...
  14, 28, 70, 140, 238, 364, 518, 700, 910, 1148, 1414, 1708, 2030, 2380, 2758, ...
  15, 30, 75, 150, 255, 390, 555, 750, 975, 1230, 1515, 1830, 2175, 2550, 2955, ...
See the attached text file for further examples.
See A357007, A357060, A357197 for more images of the n-gons.

Scott R. Shannon, Extended table for n = 3..50, k = 0..75.
Scott R. Shannon, Image of T(5,20) = 2005.
Scott R. Shannon, Image of T(7,10) = 707.

Cf. A357216 (regions), A357254 (edges), A357007 (triangle), A357060 (square), A357197 (hexagon), A007569, A146212.

T(n,k) = A357254(n,k) - A357216(n,k) + 1 by Euler's formula.
T(n,0) = n.
T(n,1) = 2n.
Conjectured formula for all columns for n >= 7: T(n,k) = n*k^2 + n.
T(3,k) = A357007(k).
T(4,k) = A357060(k).
T(6,k) = A357197(k).
Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = n*k^2 + n.

A211383 Number of intersections of diagonals in the interior and exterior of a regular n-gon.

Original entry on oeis.org

0, 1, 5, 13, 42, 73, 189, 271, 572, 661, 1365, 1569, 2790, 3057, 5117, 4555, 8664, 9041, 13797, 14213, 20930, 18625, 30525, 30967, 43092, 43513, 59189, 45871, 79422, 79713, 104445, 104619, 134960, 124921, 171717, 171533, 215514, 215081, 267197, 234319, 327660, 326569, 397845, 396337

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/9508209 [math.MG], 1995-2006.
Eric Weisstein's World of Mathematics, Regular Polygon Division by Diagonals

Cf. A006561, A146212, A211382.

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+j):
                    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 (l!=n+i) and (not y.is_zero()): S.add(x/y)
    return len(S)  # Robin Visser, Jul 29 2024

a(n) = (1/8)*n*(n-3)*(n^2-8*n+19) for n odd.

a(n) = A006561(n) + A211382(n).

More terms from Robin Visser, Jul 29 2024

A351924 The number of vertices on a diagonal of a regular 2n-gon when all its vertices are connected by lines and where the diagonal passes through the center of the 2n-gon.

Original entry on oeis.org

3, 5, 7, 11, 15, 21, 27, 29, 43, 53, 59, 75, 87, 85, 115, 131, 135, 165, 183, 185, 223, 245, 251, 291, 315, 317, 367, 395, 379, 453, 483, 485, 547, 581, 587, 651, 687, 689, 763, 803, 795, 885, 927, 925, 1015, 1061, 1067, 1155, 1203, 1205, 1303, 1355, 1359, 1461, 1515, 1517, 1627, 1685, 1659

Offset: 2

Scott R. Shannon, Feb 25 2022

nonn

No formula for a(n) is currently known.

			a(3) = 5 as a diagonal of a 6-gon, when all its vertices are connected by lines and where the diagonal passes through the center of the 6-gon, has five vertices on it - the two outer vertices, the central vertex, and two more vertices formed by intersections of the central diagonal with other diagonals. See the linked image of the 6-gon.

Scott R. Shannon, Image of the 6-gon.
Scott R. Shannon, Image of the 10-gon.
Scott R. Shannon, Image of the 18-gon.
Scott R. Shannon, Image of the 30-gon.

Cf. A007569, A006561, A146212.

A383461 Number of vertices in graph G_n formed by taking a regular n-gon with all its chords extended to infinity (the n-th graph in A344857) and inverting it in its circumscribing circle.

Original entry on oeis.org

4, 5, 16, 37, 92, 145, 334, 471, 892, 901, 1964, 2185, 3796, 3969, 6682, 5563, 10964, 11141, 17032, 17293, 25324, 21913, 36326, 36479, 50572, 50485, 68644, 51661, 91172, 90753, 118834, 118355, 152356, 139861, 192512, 191445, 240124, 238481

Offset: 3

Scott R. Shannon and N. J. A. Sloane, Jun 02 2025

nonn

Inverting a point or a line in a circle C with center O and radius r is a classical operation in geometry (Coxeter, Section 6.3; Pedoe, pp. 4-9). Every point A inside C except O itself has an inverse point A' outside the circle; A' lies on the line OA and satisfies |OA|*|OA'| = r^2. The inverse of the center O is undefined.
If a line L passes through O its inverse is L itself. If L is not a diameter of C, and meets C in two points A and B, the inverse of L is the circle through O, A, and B.
Theorem: G_n has A345025(n) regions. If n is even then n of these regions are infinite, otherwise there is a single infinite region.
The initial versions of the illustrations were made by NJAS using GeoGebra. The colored versions were added later by SRS using a Java program. These have greater resolution and include information about the vertex and region counts.

H. S. M. Coxeter, Introduction to Geometry, Wiley, 1961.
D. Pedoe, Circles: A Mathematical View, Dover, 1979.

Scott R. Shannon, The graph G_13 consists of 78 circles. There are a(13) = 1964 vertices.
Scott R. Shannon, The graph G_13 (continued). There are 2172 regions (2171 finite regions and one infinite region).
N. J. A. Sloane, The graph G_3 consists of three circles. There are a(3) = 4 vertices and 7 regions (6 finite regions and one infinite region).
N. J. A. Sloane, GeoGebra source file for G_3
N. J. A. Sloane, The graph G_4 consists of two lines and four circles. There are a(4) = 5 vertices and 16 regions (12 finite regions and 4 infinite regions).
N. J. A. Sloane, GeoGebra source file for G_4
N. J. A. Sloane, The graph G_5 consists of ten circles. There are a(5) = 16 vertices and 36 regions (35 finite regions and one infinite region).
N. J. A. Sloane, GeoGebra source file for G_5
N. J. A. Sloane, The graph G_6 consists of three lines and 12 circles. There are a(6) = 37 vertices and 72 regions (66 finite regions and 6 infinite regions).
N. J. A. Sloane, GeoGebra source file for G_6
N. J. A. Sloane, The graph G_7 consists of 21 circles (colored red). There are a(7) = 92 vertices and 141 regions (140 finite regions and one infinite region). See following illustration for an enlargement of the central heptagonal portion of the graph. Note that the blue heptagon is not part of G_7.
N. J. A. Sloane, An enlargement of the central heptagonal portion of the previous illustration, showing the 91 individual cells more clearly.
N. J. A. Sloane, GeoGebra source file for G_7
N. J. A. Sloane, The graph G_8 consists of four lines and 24 circles. There are 145 vertices and 232 regions (224 finite regions and 8 infinite regions).
N. J. A. Sloane, GeoGebra source file for G_8

Cf. A146212, A344857, A345025, A383462.

a(n) = A146212(n) + (n mod 2).

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.

A352434 The number of simple vertices on a diagonal of a regular 2n-gon when all its vertices are connected by lines and where the diagonal passes through the center of the 2n-gon.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 6, 6, 8, 8, 10, 8, 12, 12, 14, 14, 16, 16, 18, 18, 20, 20, 22, 20, 24, 24, 26, 26, 28, 28, 30, 30, 32, 32, 34, 32, 36, 36, 38, 38, 40, 40, 42, 42, 44, 44, 46, 44, 48, 48, 50, 50, 52, 52, 54, 54, 56, 56, 58, 56, 60, 60, 62, 62, 64, 64, 66, 66, 68, 68, 70, 68, 72, 72, 74, 74, 76

Offset: 1

Scott R. Shannon, Mar 16 2022

nonn

Excluding a(2), which has its simple vertex at the center of the 4-gon, the terms predominantly follow a pattern of pairs of two equal numbers and where the pair values increment by two. The second term of each pair corresponds to 2n-gons where n is a multiple of 2. These 2n-gons have two vertices that are on the same horizontal line as the central non-simple vertex thus the line joining them will not form a new simple vertex with the central vertical diagonal. Therefore in general a(2*k) = a(2*k-1), k>=1. However this rule is broken when n is a multiple of 12 - for these 2n-gons two of the horizontal lines connecting the left-side and right-side vertices also intersect two non-central diagonals and thus two simple vertices are removed. See the linked image of the 24-gon.

			a(2) = 1 as the 4-gon (square) has one simple vertex at its center when all its vertices are connected by lines.
a(3) = 2 as the 6-gon (hexagon) has two simple vertices along the central diagonal when its vertices are connected by lines. See the linked image.
a(7) = 6 as the 14-gon has six simple vertices along the central diagonal when its vertices are connected by lines. See the linked image.

Scott R. Shannon, Image of the 6-gon.
Scott R. Shannon, Image of the 10-gon.
Scott R. Shannon, Image of the 14-gon.
Scott R. Shannon, Image of the 24-gon.

Cf. A351924 (all vertices on diagonal), A352144 (all simple vertices), A292104, A007569, A006561, A146212.

cp's OEIS Frontend

A344907 Number of polygon edges formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

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A347319 a(n) = (2*n+1)*(n^3-2*n^2+n+1).

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A357235 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of vertices in an n-gon when k internal n-gons are drawn between the n*k points that divide each side into k+1 equal parts.

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A211383 Number of intersections of diagonals in the interior and exterior of a regular n-gon.

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A351924 The number of vertices on a diagonal of a regular 2n-gon when all its vertices are connected by lines and where the diagonal passes through the center of the 2n-gon.

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A383461 Number of vertices in graph G_n formed by taking a regular n-gon with all its chords extended to infinity (the n-th graph in A344857) and inverting it in its circumscribing circle.

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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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A352434 The number of simple vertices on a diagonal of a regular 2n-gon when all its vertices are connected by lines and where the diagonal passes through the center of the 2n-gon.

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A347319 a(n) = (2n+1)(n^3-2*n^2+n+1).