A344279 -id:A344279 - cp's OEIS frontend

A344657 Number of vertices formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

3, 10, 85, 310, 999, 2299, 4674, 8878, 14539, 23116, 35922, 53830, 74685, 106957, 140887, 183718, 240108, 315997, 392049, 497518, 599596, 730762, 888903, 1083277, 1257270, 1502830, 1760371, 2047792, 2362620, 2771437, 3129933, 3629443, 4107994, 4670305, 5245447, 5921755

Offset: 1

Scott R. Shannon, Jun 24 2021

nonn

See A344279 for other images of the polygons.

Scott R. Shannon, Image of the vertices for n = 3.
Scott R. Shannon, Image of the vertices for n = 4.

Cf. A344279 (number of polygons), A344896 (number of edges), A092867 (number polygons inside the triangle), A346446 (number of k-gons), A345649, A146212.

a(n) = A344896(n) - A344279(n) + 1.

A344896 Number of polygon edges 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

3, 21, 186, 705, 2196, 5046, 10212, 19149, 31524, 50049, 77100, 114633, 159453, 225978, 298833, 390069, 508137, 663426, 824868, 1040727, 1258833, 1532565, 1859331, 2254938, 2628309, 3130227, 3664920, 4261503, 4917939, 5742696, 6503331, 7511280, 8507322, 9658806, 10855989, 12237066

Offset: 1

Scott R. Shannon, Jun 24 2021

nonn

See A344279 for images of the polygons.

Cf. A344279 (number of polygons), A344657 (number of vertices), A346446 (number of k-gons), A092867 (number polygons inside the triangle), A344899, A345650.

a(n) = A344279(n) + A344657(n) - 1.

A346446 Irregular triangle read by rows: T(n,k) = number of k-sided polygons formed when connecting infinite lines between all vertices and all points that divide the sides of an equilateral triangle into n equal parts, for k = 3, 4, ..., max_k.

Original entry on oeis.org

1, 12, 75, 24, 3, 258, 132, 6, 621, 525, 33, 19, 1308, 1272, 144, 24, 2505, 2628, 345, 61, 4434, 4734, 984, 102, 12, 6, 7365, 7992, 1347, 243, 30, 9, 11556, 12552, 2412, 366, 48, 17073, 19266, 3969, 804, 60, 3, 0, 3, 0, 1, 24786, 27672, 6954, 1206, 186, 34611, 39066, 9099, 1768, 198, 27

Offset: 1

Scott R. Shannon, Jul 18 2021

See A344279 for other images of the polygons.

			Connecting infinite lines between an equilateral triangle's three vertices and the two points along each side that divide the sides into three equal parts forms seventy-five triangles, twenty-four quadrilaterals and three pentagons, so row 3 is [75,24,3]. See the linked image.
The table begins:
       1;
      12;
      75,     24,      3;
     258,    132,      6;
     621,    525,     33,    19;
    1308,   1272,    144,    24;
    2505,   2628,    345,    61;
    4434,   4734,    984,   102,   12,   6;
    7365,   7992,   1347,   243,   30,   9;
   11556,  12552,   2412,   366,   48;
   17073,  19266,   3969,   804,   60,   3,  0,  3, 0, 1;
   24786,  27672,   6954,  1206,  186;
   34611,  39066,   9099,  1768,  198,  27;
   47028,  53688,  15318,  2676,  288,  24;
   63039,  72210,  18513,  3708,  396,  75,  0,  6;
   82746,  93570,  24930,  4536,  498,  54, 18;
  106536, 121080,  32988,  6622,  678, 117,  6,  3;
  134520, 155748,  46326,  9456, 1266, 102, 12;
  167895, 196179,  55527, 11410, 1638, 156, 12,  3;
  207294, 243294,  74796, 15396, 2106, 276, 42,  6;
  254034, 297069,  87648, 17715, 2388, 363, 18,  3;
  308022, 360228, 108264, 21858, 3090, 282, 42, 18;
  370818, 433902, 132651, 28210, 4311, 486, 42,  9;
  440952, 520044, 168156, 36228, 5484, 720, 78;
  521031, 614526, 189297, 39541, 5790, 780, 60, 15;
  612990, 723228, 232980, 49278, 8004, 822, 96;

Scott R. Shannon, Image of the k-gons for n=3.

Cf. A344279 (number of polygons), A344657 (number of vertices), A344896 (number of edges), A343755 (number of regions), A092867 (number polygons inside the triangle).

Sum of row(n) = A344279(n) = A344896(n) - A344657(n) + 1.

A343755 Number of regions 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

7, 30, 144, 474, 1324, 2934, 5797, 10614, 17424, 27480, 41845, 61602, 85711, 120120, 159213, 207798, 269668, 349272, 434878, 545496, 661764, 804582, 973471, 1174980, 1374646, 1631304, 1908768, 2218254, 2560198, 2976486, 3378985, 3887796, 4405671, 4995240, 5617689, 6322878

Offset: 1

Scott R. Shannon, Jun 28 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. The number of unbounded areas appears to equal 6*(n^2 - n + 1).

See A344279 for further examples and images of the regions.

			a(1) = 7 as the three connected vertices of a triangle form one polygon along with six outer unbounded areas, seven regions in total.
a(2) = 30 as when the three vertices and three edges points are connected they form twelve polygons, all inside the triangle, along with eighteen outer unbounded areas, thirty regions in total.
a(2) = 144 as when the three vertices and six edges points are connected they form one hundred two polygons, seventy-five inside the triangle and twenty-seven outside, along with forty-two outer unbounded areas, one hundred forty-four regions in total.

Scott R. Shannon, Image for n = 1. In this and other images the triangle's 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 = 2.
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. A344279 (number of polygons), A344657 (number of vertices), A344896 (number of edges), A346446 (number of k-gons), A092867 (number polygons inside the triangle), A121205, A345025.

Conjectured formula: a(n) = A344279(n) + 6*(n^2 - n + 1).

Conjectured formula: a(n) = A344279(n) + A121205(n-1), for n>=7.

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.

cp's OEIS Frontend

A344657 Number of vertices formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

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A344896 Number of polygon edges formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

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A346446 Irregular triangle read by rows: T(n,k) = number of k-sided polygons formed when connecting infinite lines between all vertices and all points that divide the sides of an equilateral triangle into n equal parts, for k = 3, 4, ..., max_k.

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A343755 Number of regions formed by infinite lines when connecting all vertices and all points that divide the sides of an equilateral triangle into n equal parts.

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Examples

Links

Crossrefs

Formula

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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Author

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