A344857 -id:A344857 - cp's OEIS frontend

A347320 A344857(2*n).

0, 4, 42, 176, 540, 1152, 2534, 4608, 6984, 12580, 19250, 26016, 39988, 55076, 63570, 97920, 127024, 155088, 203946, 253360, 289044, 378884, 456918, 528672, 648200, 763620, 869022, 1040368, 1204196, 1318440, 1588750, 1812224, 2013132, 2329476, 2626330, 2910672, 3303804, 3687748, 4039854, 4555520, 5043000, 5471088, 6133778, 6740976, 7295760, 8090020, 8836094, 9544128, 10481296, 11384900

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Sep 12 2021, extended Jun 23 2025

No formula is known.

Cf. A344857, A344866.

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).

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

Original entry on oeis.org

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

A358782 The number of regions formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

Original entry on oeis.org

1, 7, 12, 66, 85, 281, 264, 802, 821, 1893, 1740, 3810, 3725, 6871, 6448, 11748, 11125, 18317, 17160, 27616, 26797, 40067, 37176, 56826, 54653, 77707, 74788, 103734, 101041, 136835, 131744, 176584, 172109, 223931, 216900, 281090, 273829, 348583, 337480, 425950, 416641

Offset: 2

Scott R. Shannon, Nov 30 2022

nonn

Conjecture: for odd values of n all vertices are simple, other than those defining the diameters of the circles. No formula for n, or only the odd values of n, is currently known.
The author thanks Zach Shannon some of whose code was used in the generation of this sequence.
If n is odd, the circle containing the initial n points is not part of the graph (compare A370976-A370979). - N. J. A. Sloane, Mar 25 2024

Scott R. Shannon, Image for n = 2. In this and other images the points defining the circle diameters are show as white dots.
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.
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.
Scott R. Shannon, Image for n = 17.
Scott R. Shannon, Image for n = 20.
Scott R. Shannon, Image for n = 23.

Cf. A358746 (vertices), A358783 (edges), A359009 (k-gons), A007678, A344857.

See allso A370976-A370979.

a(n) = A358783(n) - A358746(n) + 1 by Euler's formula.

A344899 Number of polygon edges formed when every pair of vertices of a regular n-gon are joined by an infinite line.

Original entry on oeis.org

0, 1, 3, 8, 30, 78, 189, 320, 684, 1010, 1815, 2052, 3978, 4718, 7665, 8576, 13464, 12546, 22059, 23720, 34230, 36542, 50853, 47928, 72900, 76466, 101439, 105560, 137634, 115230, 182745, 188672, 238128, 245378, 305235, 294948, 385614, 395390, 480909, 491840, 592860, 544950, 723303, 737528

Offset: 1

Scott R. Shannon, Jun 02 2021

nonn

See A344857 for other examples and images of the polygons.

			a(3) = 3 as the connected vertices form a triangle with three edges. Six infinite edges between the outer regions are also formed but these are not counted.
a(5) = 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 regions are also formed.

J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.
Alexander Sidorenko, Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857.

Cf. A344907 (number of edges for odd n), A344857 (number of polygons), A146212 (number of vertices), A344866, A344311, A007678, A331450, A344938.

Bisections: A344907, A347322.

Conjectured formula odd n: a(n) = (n^4 - 7*n^3 + 17*n^2 - 11*n)/4 = (n-1)*n*(n^2-6*n+11)/4.
This formula is correct: see the Sidorenko link. - N. J. A. Sloane, Sep 12 2021
See also A344907.
a(n) = A344857(n) + A146212(n) - 1 (Euler's theorem.).

A344866 Number of polygons formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

Original entry on oeis.org

0, 1, 16, 99, 352, 925, 2016, 3871, 6784, 11097, 17200, 25531, 36576, 50869, 68992, 91575, 119296, 152881, 193104, 240787, 296800, 362061, 437536, 524239, 623232, 735625, 862576, 1005291, 1165024, 1343077, 1540800, 1759591, 2000896, 2266209, 2557072, 2875075, 3221856, 3599101, 4008544, 4451967

Offset: 1

Scott R. Shannon, Jun 01 2021

This is the odd-indexed subsequence of A344857. See A344857 for images of the polygons.

			a(3) = 16 as the five connected vertices form eleven polygons inside the regular pentagon while also forming five triangles outside the pentagon, giving sixteen polygons in total.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
John Elias, Illustration of initial terms: Cubic Vertex Fractal
J. F. Rigby, Multiple intersections of diagonals of regular polygons, and related topics, Geom. Dedicata 9 (1980), 207-238.
Alexander Sidorenko, Explicit Formulas for Odd-Indexed Terms in A344899, A146212, and A344857.
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A344857 (number for even and odd n), A344311, A344938, A007678, A341735 (number inside the n-gon), A344899 (number of edges).

A344938 Irregular triangle read by rows: T(n,k) = number of k-sided polygons formed when every pair of vertices of a regular n-gon are joined by an infinite line, for k = 3, 4, ..., max_k.

Original entry on oeis.org

1, 4, 15, 0, 1, 36, 6, 70, 21, 7, 0, 1, 112, 64, 189, 108, 36, 18, 0, 0, 1, 270, 220, 50, 407, 352, 110, 55, 0, 0, 0, 0, 1, 624, 528, 884, 689, 325, 91, 0, 26, 0, 0, 0, 0, 1, 1162, 1092, 266, 14, 1530, 1545, 480, 270, 45, 0, 0, 0, 0, 0, 0, 0, 1, 2080, 2032, 416, 80

Offset: 3

Scott R. Shannon, Jun 03 2021

See A344857 for examples and images of the polygons.

			A pentagon with all vertices connected forms 10 triangles inside the pentagon, 5 triangles outside the pentagon, giving 15 triangles in all, and 1 smaller pentagon inside the pentagon, so row 3 is [15,0,1].
The table begins:
1;
4;
15,0,1;
36,6;
70,21,7,0,1;
112,64;
189,108,36,18,0,0,1;
270,220,50;
407,352,110,55,0,0,0,0,1;
624,528;
884,689,325,91,0,26,0,0,0,0,1;
1162,1092,266,14;
1530,1545,480,270,45,0,0,0,0,0,0,0,1;
2080,2032,416,80;
2567,2754,1003,374,17,68,0,0,0,0,0,0,0,0,1;
3402,3366,180,18,18;
3952,4807,1672,475,95,76,0,19,0,0,0,0,0,0,0,0,1;
5380,5360,1580,240,0,20;
5943,7392,2583,1260,21,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
7590,9020,2310,132,132,66;
9430,9775,4508,1518,253,46,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
11304,12288,2280,144;
13025,14650,6250,2375,200,75,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
16042,16952,5954,728,260,52;
17064,22464,7884,2700,567,189,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
21616,24192,7056,2016,168,28;
23751,29319,11281,3828,348,319,0,87,29,29,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
29880,29010,4140,540;
30814,39370,15314,5177,341,496,0,62,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;
37440,42624,14240,3008,544,64;
41481,49335,19305,7854,891,363,66,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1;

Cf. A344857 (total number of polygons), A344899 (number of edges), A146212 (number of vertices), A344866, A344311, A007678, A331450 (number of k-gons inside the regular n-gon).

Sum of row(n) = A344857(n) = A344311(n) + A007678(n).

A345459 Number of polygons formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

Original entry on oeis.org

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

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jun 20 2021

The width/height of the entire figure grows as ~ 2*n^3 for large n. See the Formula section below.

			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.

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.

Cf. A255011 (number inside the square), A345648 (number outside the square), A345649 (number of vertices), A345650 (number of edges), A344993, A344857, A092098, A007678.

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).

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

A344279 Number of polygons formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

Original entry on oeis.org

1, 12, 102, 396, 1198, 2748, 5539, 10272, 16986, 26934, 41179, 60804, 84769, 119022, 157947, 206352, 268030, 347430, 432820, 543210, 659238, 801804, 970429, 1171662, 1371040, 1627398, 1904550, 2213712, 2555320, 2971260, 3373399, 3881838, 4399329, 4988502, 5610543, 6315312

Offset: 1

Scott R. Shannon, Jun 22 2021

nonn

Scott R. Shannon, Image for n = 2. In this and other images the triangle'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.
Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.

Cf. A344657 (number of vertices), A344896 (number of edges), A346446 (number of k-gons), A092867 (number polygons inside the triangle), A343755 (number of regions), A345459, A344857.

a(n) = A344896(n) - A344657(n) + 1.

cp's OEIS Frontend

A347320 A344857(2*n).

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

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A358782 The number of regions formed when every pair of n points, placed at the vertices of a regular n-gon, are connected by a circle and where the points lie at the ends of the circle's diameter.

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A344899 Number of polygon edges formed when every pair of vertices of a regular n-gon are joined by an infinite line.

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A344866 Number of polygons formed when every pair of vertices of a regular (2n-1)-gon are joined by an infinite line.

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A344938 Irregular triangle read by rows: T(n,k) = number of k-sided polygons formed when every pair of vertices of a regular n-gon are joined by an infinite line, for k = 3, 4, ..., max_k.

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A345459 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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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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