A344857 -id:A344857 - cp's OEIS frontend

A345025 Number of regions formed when every pair of vertices of a regular n-gon are joined by an infinite line.

1, 2, 7, 16, 36, 72, 141, 232, 424, 630, 1035, 1284, 2172, 2716, 4081, 4848, 7056, 7290, 11439, 12960, 17620, 19712, 26037, 26568, 37176, 40638, 51571, 55832, 69804, 64440, 92505, 98912, 120352, 128146, 154071, 156348, 194436, 205352, 242269, 254920, 298440, 290766, 363867, 380776, 439516

Offset: 1

Scott R. Shannon, Jun 06 2021

nonn

The count of regions includes both the closed bounded polygons and the open unbounded areas surrounding these polygons with two edges that go to infinity.

See A344857 for further examples and images of the regions.

			a(2) = 2 as an infinite line connecting two points cuts space into two unbounded regions.
a(3) = 7 as the three connected points of the 3-gon form one closed triangle along with six outer unbounded areas, seven regions in total.
a(4) = 16 as the four connected points of the 4-gon form four closed triangle inside the square along with twelve outer unbounded areas, sixteen regions in total.

Scott R. Shannon, Image for n = 3. In this and other images the n-gon vertices are highlighted as white dots while the outer open regions are cross-hatched. The key for the edge-number coloring is shown at the top-left of the image. Note the edge count for open areas also includes the two infinite edges.
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.

Cf. A344857 (number of polygons), A344311 (number polygons outside the n-gon), A007678 (number polygons inside the n-gon), A002378 (number of open regions for (n-1)-gon), A146212 (number of vertices), A344866, A344938.

Formula for odd n: a(n) = (n^4 - 7*n^3 + 27*n^2 - 29*n + 8)/8 (see A344857).

For n >= 3, a(n) = A344857(n) + A002378(n-1).

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

Original entry on oeis.org

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.

A345649 Number of vertices formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

Original entry on oeis.org

0, 5, 61, 465, 1585, 5257, 9749, 24025, 38381, 67177, 100889, 176005, 210033, 360877, 450349, 589581, 779541, 1127509, 1251805, 1806061, 1970129, 2504401, 3116945, 4017701, 4163753, 5433657, 6335589

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jun 21 2021

See A345459 for images of the polygons.

Cf. A345459 (number of polygons), A345650 (number of edges), A255011, A345648, A344993, A344857, A092098, A007678.

a(n) = A345650(n) - A345459(n) + 1.

A345650 Number of polygon edges formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

Original entry on oeis.org

0, 8, 140, 1032, 3608, 11308, 21892, 51000, 83404, 143900, 217728, 367132, 456008, 749328, 952236, 1251056, 1649708, 2327232, 2653900, 3717444, 4158448, 5243680, 6488208, 8241988, 8780976, 11235028, 13116156

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jun 21 2021

See A345459 for images of the polygons.

Cf. A331448 (number inside the square), A345459 (number of polygons), A345649 (number of vertices), A255011, A345648, A344993, A344857, A092098, A007678.

a(n) = A345459(n) + A345649(n) - 1.

A386560 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join an infinite straight line between every pair of points: a(n) is the number of finite regions created in the resulting graph.

Original entry on oeis.org

4, 72, 424, 1396, 3536, 7292, 14272, 24332, 39356, 59920, 91348, 128084, 182664, 245804, 323116, 418552, 547820, 684680, 869388, 1060892, 1289564, 1560920

Offset: 1

Scott R. Shannon, Jul 26 2025

Scott R. Shannon, Image for n = 1. In this and other images the integer coordinates are highlighted as white dots while the outer open regions, which are not counted, are darkened.
Scott R. Shannon, Image for n = 2.
Scott R. Shannon, Image for n = 3.
Scott R. Shannon, Image for n = 4.
Scott R. Shannon, Image for n = 5

Cf. A386559 (vertices), A386561 (edges), A386562 (k-gons), A344993, A344857, A344279, A345459.

a(n) = A386561(n) - A386559(n) + 1 by Euler's formula.

A345648 Number of polygons formed outside an n X n square when connecting all 4n points on the perimeter of the square by infinite lines.

Original entry on oeis.org

0, 0, 24, 228, 904, 2788, 5880, 13008, 22120, 37976, 58584, 95472, 125016, 195816, 255064, 337916, 444760, 611760, 719800, 978388, 1127088, 1411756, 1736776, 2174584, 2389552, 2995336, 3504768

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jun 21 2021

For n = 0 to n = 11 the number of polygons formed outside the n x n square is less than the number formed inside the square. This changes for n >= 12 when the number formed outside becomes greater. The ratio of the number of polygons outside to the number inside for n = 26 is about 1.07 . If this ratio is unbounded or approaches some finite value as n -> infinity is not known.

See A345459 for images of the polygons.

Cf. A345459 (number inside and outside with square), A255011 (number inside the square), A344993, A344857, A092098, A007678.

a(n) = A345459(n) - A255011(n).

A352533 Irregular table read by rows: T(n,k) is the number of regions formed after k diagonals, with k>=0, are drawn between vertices of a regular n-gon, with n>=3, when each vertex is fully connected to all other vertices in counterclockwise order before the next vertex, in counterclockwise order, is chosen.

Original entry on oeis.org

1, 1, 2, 4, 1, 2, 3, 5, 8, 11, 1, 2, 3, 4, 6, 9, 13, 16, 20, 24, 1, 2, 3, 4, 5, 7, 10, 14, 19, 22, 27, 34, 38, 45, 50, 1, 2, 3, 4, 5, 6, 8, 11, 15, 20, 26, 29, 34, 39, 46, 50, 56, 62, 67, 74, 80, 1, 2, 3, 4, 5, 6, 7, 9, 12, 16, 21, 27, 34, 37, 42, 49, 58, 69, 73, 80, 90, 103, 108, 117, 130, 136, 147, 154

Offset: 3

Scott R. Shannon, Mar 19 2022

To create the diagonals between the vertices of the regular n-gon a random starting vertex is first chosen. This vertex is then connected to all other vertices where those vertices are chosen in a counterclockwise direction around the n-gon from the starting vertex. Once all those diagonals are drawn the next neighboring vertex, in a counterclockwise direction from the starting vertex, is chosen. This vertex is then connected to all other vertices in the same way. This method is repeated until all vertices are connected by diagonals. The sequence gives the number of regions inside the n-gon after each such diagonal is drawn.

			The table begins:
1;
1,2,4;
1,2,3,5,8,11;
1,2,3,4,6,9,13,16,20,24;
1,2,3,4,5,7,10,14,19,22,27,34,38,45,50;
1,2,3,4,5,6,8,11,15,20,26,29,34,39,46,50,56,62,67,74,80;
1,2,3,4,5,6,7,9,12,16,21,27,34,37,42,49,58,69,73,80,90,103,108,117,130,136,147,154;
1,2,3,4,5,6,7,8,10,13,17,22,28,35,43,46,51,58,65,75,86,90,97,107,116,130,135,143, \
      154,164,170,180,194,201,212,220;
1,2,3,4,5,6,7,8,9,11,14,18,23,29,36,44,53,56,61,68,77,88,101,116,120,127,137,150, \
      166,185,190,199,212,229,250,256,267,283,304,311,324,343,351,366,375;
.
.
See the linked file for the table up to n=100. See the linked images for examples of the 7-gon.

Scott R. Shannon, Table for n=3..100.
Scott R. Shannon, Image for the 7-gon after first vertex is connected to all other vertices. Total regions = 5.
Scott R. Shannon, Image for the 7-gon after second vertex is connected to all other vertices. Total regions = 19.
Scott R. Shannon, Image for the 7-gon after third vertex is connected to all other vertices. Total regions = 34.
Scott R. Shannon, Image for the 7-gon after fourth vertex is connected to all other vertices. Total regions = 45.
Scott R. Shannon, Image for the 7-gon after fifth vertex is connected to all other vertices. Total regions = 50.

Cf. A007678, A350000, A344857.

The last term in each row n = A007678(n).

A357216 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of regions 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

1, 4, 1, 13, 5, 1, 28, 17, 6, 1, 49, 37, 21, 7, 1, 70, 65, 46, 25, 8, 1, 109, 93, 81, 55, 29, 9, 1, 148, 145, 126, 97, 64, 33, 10, 1, 181, 181, 181, 151, 113, 73, 37, 11, 1, 244, 257, 246, 217, 176, 129, 82, 41, 12, 1, 301, 309, 321, 295, 253, 201, 145, 91, 45, 13, 1

Offset: 3

Scott R. Shannon, Sep 18 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:
  1,  4, 13,  28,  49,  70, 109, 148, 181,  244,  301,  334,  433,  508,  565, ...
  1,  5, 17,  37,  65,  93, 145, 181, 257,  309,  401,  457,  577,  653,  785, ...
  1,  6, 21,  46,  81, 126, 181, 246, 321,  406,  501,  606,  721,  846,  981, ...
  1,  7, 25,  55,  97, 151, 217, 295, 385,  475,  601,  715,  865, 1015, 1159, ...
  1,  8, 29,  64, 113, 176, 253, 344, 449,  568,  701,  848, 1009, 1184, 1373, ...
  1,  9, 33,  73, 129, 201, 289, 393, 513,  649,  801,  969, 1153, 1353, 1569, ...
  1, 10, 37,  82, 145, 226, 325, 442, 577,  730,  901, 1090, 1297, 1522, 1765, ...
  1, 11, 41,  91, 161, 251, 361, 491, 641,  811, 1001, 1211, 1441, 1691, 1961, ...
  1, 12, 45, 100, 177, 276, 397, 540, 705,  892, 1101, 1332, 1585, 1860, 2157, ...
  1, 13, 49, 109, 193, 301, 433, 589, 769,  973, 1201, 1453, 1729, 2029, 2353, ...
  1, 14, 53, 118, 209, 326, 469, 638, 833, 1054, 1301, 1574, 1873, 2198, 2549, ...
  1, 15, 57, 127, 225, 351, 505, 687, 897, 1135, 1401, 1695, 2017, 2367, 2745, ...
  1, 16, 61, 136, 241, 376, 541, 736, 961, 1216, 1501, 1816, 2161, 2536, 2941, ...
  ...
See the attached text file for further examples.
See A356984, A357058, A357196 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) = 2001.
Scott R. Shannon, Image of T(7,10) = 701.

Cf. A357235 (vertices), A357254 (edges), A356984 (triangle), A357058 (square), A357196 (hexagon), A007678, A344857.

T(n,k) = A357254(n,k) - A357235(n,k) + 1 by Euler's formula.
T(n,0) = 1.
T(n,1) = n + 1.
Conjectured formula for all columns for n >= 7: T(n,k) = n*k^2 + 1.
T(3,k) = A356984(k).
T(4,k) = A357058(k).
T(6,k) = A357196(k).
Conjectured formula for all rows except for n = 3, 4, 6: T(n,k) = n*k^2 + 1.

A352866 Irregular table read by rows: T(n,k) is the number of regions formed after k diagonals, with k>=0, are drawn between vertices of a regular n-gon, with n>=3, when each vertex in turn is connected to the vertex two to its left, then three to its left, then four... until all vertices are connected by diagonals.

Original entry on oeis.org

1, 1, 2, 4, 1, 2, 4, 6, 8, 11, 1, 2, 4, 6, 8, 10, 13, 16, 20, 24, 1, 2, 4, 6, 8, 10, 12, 15, 18, 22, 27, 32, 37, 43, 50, 1, 2, 4, 6, 8, 10, 12, 14, 17, 20, 24, 29, 34, 39, 44, 50, 57, 62, 68, 74, 80, 1, 2, 4, 6, 8, 10, 12, 14, 16, 19, 22, 26, 31, 36, 41, 46, 51, 57, 64, 71, 79, 88, 98, 108, 118, 129, 141, 154

Offset: 3

Scott R. Shannon, Apr 06 2022

To create the diagonals between the vertices of the regular n-gon a random starting vertex is first chosen. This vertex is then connected to the vertex two to its left. The left neighboring vertex of the starting vertex is then chosen and this is connected to the vertex two to its left. This process is continued until all vertices are connected by diagonals to the vertices two to their left. The initial vertex is then chosen again and it is connected to the vertex three to its left. Its left neighboring vertex is then connected to the vertex three to its left, and so on. This process of connecting all vertices to those on their left by diagonals, where the step size increases by one after each complete circuit of the n-gon, is continued until all vertices are connected by diagonals. The sequence gives the number of regions inside the n-gon after each such diagonal is drawn.

			The table begins:
1;
1,2,4;
1,2,4,6,8,11;
1,2,4,6,8,10,13,16,20,24;
1,2,4,6,8,10,12,15,18,22,27,32,37,43,50;
1,2,4,6,8,10,12,14,17,20,24,29,34,39,44,50,57,62,68,74,80;
1,2,4,6,8,10,12,14,16,19,22,26,31,36,41,46,51,57,64,71,79,88,98,108,118,129, \
        141,154;
1,2,4,6,8,10,12,14,16,18,21,24,28,33,38,43,48,53,58,64,71,78,86,95,105,115, \
        125,135,146,158,171,180,190,200,210,220;
1,2,4,6,8,10,12,14,16,18,20,23,26,30,35,40,45,50,55,60,65,71,78,85,93,102,112, \
        122,132,142,152,163,175,188,201,215,230,246,263,280,297,315,334,354,375;
.
.
See the linked file for the table up to n=100. See the linked images for examples of the 10-gon.

Scott R. Shannon, Table for n=3..100.
Scott R. Shannon, Image for the 10-gon after all vertices are connected to the vertices two to their left. Total regions = 21.
Scott R. Shannon, Image for the 10-gon after all vertices are connected to the vertices three to their left. Total regions = 71.
Scott R. Shannon, Image for the 10-gon after all vertices are connected to the vertices four to their left. Total regions = 171.
Scott R. Shannon, Image for the 10-gon after all vertices are connected to the vertices five to their left. Total regions = 220.

Cf. A352533, A007678, A350000, A344857.

The last term in each row n = A007678(n).

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

A345025 Number of regions formed when every pair of vertices of a regular n-gon are joined by an infinite line.

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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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A345649 Number of vertices formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

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A345650 Number of polygon edges formed when connecting all 4n points on the perimeter of an n X n square by infinite lines.

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A386560 Place a point on the integer coordinates, up to |n|, along all four axial directions on a Cartesian plane, and then join an infinite straight line between every pair of points: a(n) is the number of finite regions created in the resulting graph.

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A345648 Number of polygons formed outside an n X n square when connecting all 4n points on the perimeter of the square by infinite lines.

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A357216 Table read by antidiagonals: T(n,k) (n >= 3, k >= 0) is the number of regions 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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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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